Infinite Sequences and Series

Sequences

A sequence is an ordered list of objects and can be considered as a function whose domain is the natural numbers.

Learning Objectives

Distinguish a sequence and a set

Key Takeaways

Key Points

  • Like a set, a sequence contains members (also called elements). Unlike a set, order matters in a sequence, and the same elements can appear multiple times at different positions.
  • The terms of a sequence are commonly denoted by a single variable, say
    ana_n
    , where the index
    nn
    indicates the
    nn
    th element of the sequence.
  • Sequences whose elements are related to the previous elements in a straightforward way are often specified using recursion.


Key Terms

  • set: a collection of distinct objects, considered as an object in its own right
  • recursion: the act of defining an object (usually a function) in terms of that object itself


A sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements, or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters in a sequence, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable, totally ordered set, such as the natural numbers.

Examples:
(M,A,R,Y)(M, A, R, Y)
is a different sequence from
(A,R,M,Y)(A, R, M, Y)
. Also, the sequence
(1,1,2,3,5,8)(1, 1, 2, 3, 5, 8)
, which contains the number
11
at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers
(2,4,6,)(2, 4, 6, \cdots)
. Finite sequences are sometimes known as strings or words, and infinite sequences as streams. The empty sequence
()( \quad )
is included in most notions of sequence, but may be excluded depending on the context.

Indexing

The terms of a sequence are commonly denoted by a single variable, say
ana_n
, where the index
nn
indicates the
nn
th element of the sequence. Indexing notation is used to refer to a sequence in the abstract. It is also a natural notation for sequences whose elements are related to the index
nn
(the element's position) in a simple way. For instance, the sequence of the first 10 square numbers could be written as:

(a1,a2,,a10),ak=k2(a_1,a_2, \cdots,a_{10}), \quad a_k = k^2


This represents the sequence
(1,4,9,,100)(1,4,9, \cdots, 100)
.

Sequences can be indexed beginning and ending from any integer. The infinity symbol,
\infty
, is often used as the superscript to represent the sequence that includes all integer
kk
-values starting with a certain one. The sequence of all positive squares is then denoted as:

(ak)k=1,ak=k2\displaystyle{(a_k)_{k=1}^\infty, \quad a_k = k^2}
.

image

A Convergent Sequence: The plot of a convergent sequence (

ana_n
) is shown in blue. Visually, we can see that the sequence is converging to the limit of
00
as
nn
increases.

Specifying a Sequence by Recursion

Sequences whose elements are related to the previous elements in a straightforward way are often specified using recursion. This is in contrast to the specification of sequence elements in terms of their position. To specify a sequence by recursion requires a rule to construct each consecutive element in terms of the ones before it. In addition, enough initial elements must be specified so that new elements of the sequence can be specified by the rule.

Example

The Fibonacci sequence can be defined using a recursive rule along with two initial elements. The rule is that each element is the sum of the previous two elements, and the first two elements are
00
 and
11
:
an=an1+an2a_n = a_{n-1} + a_{n-2}
 and
a0=0,a1=1a_0 = 0, \, a_1=1
. The first ten terms of this sequence are (
0,1,1,2,3,5,8,13,21,340,1,1,2,3,5,8,13,21,34
).

Series

A series is the sum of the terms of a sequence.

Learning Objectives

State the requirements for a series to converge to a limit

Key Takeaways

Key Points

  • Given an infinite sequence of numbers
    {an}\{ a_n \}
    , a series is informally the result of adding all those terms together:
    n=0an\sum_{n=0}^\infty a_n
    .
  • Unlike finite summations, infinite series need tools from mathematical analysis, specifically the notion of limits, to be fully understood and manipulated.
  • By definition, a series converges to a limit
    LL
    if and only if the associated sequence of partial sums converges to
    LL
    :
    L=n=0anL=limkSkL = \sum_{n=0}^{\infty}a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k
    .


Key Terms

  • sequence: an ordered list of objects
  • Zeno's dichotomy: That which is in locomotion must arrive at the half-way stage before it arrives at the goal.


A series is, informally speaking, the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely. Given an infinite sequence of numbers
{an}\{ a_n \}
, a series is informally the result of adding all those terms together:
a1+a2+a3+a_1 + a_2 + a_3 + \cdots
. These can be written more compactly using the summation symbol
Σ\Sigma
. An example is the famous series from Zeno's dichotomy and its mathematical representation:

n=112n=12+14+18+\displaystyle{\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots}


image

Zeno's Paradox: Say you are working from a location

x=0x=0
toward
x=100x=100
. Before you can get there, you must get halfway there. Before you can get halfway there, you must get a quarter of the way there. Before traveling a quarter, you must travel one-eighth; before an eighth, one-sixteenth; and so on.

The terms of the series are often produced according to a certain rule, such as by a formula or by an algorithm. As there are an infinite number of terms, this notion is often called an infinite series. Unlike finite summations, infinite series need tools from mathematical analysis, specifically the notion of limits, to be fully understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, and finance.

Definition

For any sequence of rational numbers, real numbers, complex numbers, functions thereof, etc., the associated series is defined as the ordered formal sum:

n=0an=a0+a1+a2+\displaystyle{\sum_{n=0}^{\infty}a_n = a_0 + a_1 + a_2 + \cdots}


The sequence of partial sums
{Sk}\{S_k\}
 associated to a series
n=0an\sum_{n=0}^\infty a_n
 is defined for each k as the sum of the sequence
{an}\{a_n\}
 from
a0a_0
 to
aka_k
:

Sk=n=0kan=a0+a1++ak\displaystyle{S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k}


By definition, the series
n=0an\sum_{n=0}^{\infty} a_n
 converges to a limit
LL
if and only if the associated sequence of partial sums
{Sk}\{S_k\}
 converges to
LL
. This definition is usually written as follows:

L=n=0anL=limkSk\displaystyle{L = \sum_{n=0}^{\infty}a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k}


The Integral Test and Estimates of Sums

The integral test is a method of testing infinite series of nonnegative terms for convergence by comparing them to an improper integral.

Learning Objectives

Describe the purpose of the integral test

Key Takeaways

Key Points

  • The integral test uses a monotonically decreasing function
    ff
    defined on the unbounded interval
    [N,)[N, \infty)
    (where
    NN
     is an integer).
  • The infinite series
    n=Nf(n)\sum_{n=N}^\infty f(n)
    converges to a real number if and only if the improper integral
    Nf(x)dx\int_N^\infty f(x)\,dx
    is finite. In other words, if the integral diverges, then the series diverges as well.
  • Integral tests proves that the harmonic series
    n=11n\sum_{n=1}^\infty \frac1n
    diverges.


Key Terms

  • improper integral: an integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity
  • natural logarithm: the logarithm in base
    ee


The integral test for convergence is a method used to test infinite series of non-negative terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.

Statement of the test

Consider an integer
NN
 and a non-negative function
ff
defined on the unbounded interval
[N,)[N, \infty )
, on which it is monotonically decreasing. The infinite series
n=Nf(n)\sum_{n=N}^\infty f(n)
 converges to a real number if and only if the improper integral
Nf(x)dx\int_N^\infty f(x)\,dx
 is finite. In other words, if the integral diverges, then the series diverges as well.

Although we won't go into the details, the proof of the test also gives the lower and upper bounds:

Nf(x)dxn=Nf(n)f(N)+Nf(x)dx\displaystyle{\int_N^\infty f(x)\,dx\le\sum_{n=N}^\infty f(n)\le f(N)+\int_N^\infty f(x)\,dx}


for the infinite series.

Applications

The harmonic series
n=11n\sum_{n=1}^\infty \frac1n
 diverges because, using the natural logarithm (its derivative) and the fundamental theorem of calculus, we get:

1M1xdx=lnx1M=lnMfor M\displaystyle{\int_1^M\frac1x\,dx=\ln x\Bigr|_1^M=\ln M\to\infty \quad\text{for }M\to\infty}


On the other hand, the series
n=11n1+ε\sum_{n=1}^\infty \frac1{n^{1+\varepsilon}}
 converges for every
ε>0\varepsilon > 0
 because, by the power rule:

1M1x1+εdx=1εxε1M=1ε(11Mε)1ε\displaystyle{\int_1^M\frac1{x^{1+\varepsilon}}\,dx =-\frac1{\varepsilon x^\varepsilon}\biggr|_1^M= \frac1\varepsilon\Bigl(1-\frac1{M^\varepsilon}\Bigr) \le\frac1\varepsilon }


image

Integral Test: The integral test applied to the harmonic series. Since the area under the curve

y=1xy = \frac{1}{x}
 for
x[1,)x \in [1, \infty)
is infinite, the total area of the rectangles must be infinite as well.

The above examples involving the harmonic series raise the question of whether there are monotone sequences such that
f(n)f(n)
 decreases to
00
faster than
1n\frac{1}{n}
but slower than
1n1+ε\frac{1}{n^{1 + \varepsilon}}
 in the sense that:

limnf(n)1n=0\displaystyle{\lim_{n\to\infty}\frac{f(n)}{\frac{1}{n}}=0}


and:

limnf(n)1n1+ε=\displaystyle{\lim_{n\to\infty}\frac{f(n)}{\frac{1}{n^{1+\varepsilon}}}=\infty}


for every
ε>0\varepsilon > 0
, and whether the corresponding series of the
f(n)f(n)
still diverges. Once such a sequence is found, a similar question can be asked of
f(n)f(n)
 taking the role of
1n\frac{1}{n}
 oand so on. In this way, it is possible to investigate the borderline between divergence and convergence of infinite series.

Comparison Tests

Comparison test may mean either limit comparison test or direct comparison test, both of which can be used to test convergence of a series.

Learning Objectives

Distinguish the limit comparison and the direct comparison tests

Key Takeaways

Key Points

  • For sequences
    {an}\{a_n \}
    ,
    {bn}\{b_n \}
    , both with non-negative terms only, if
    limnanbn=c\lim_{n \to \infty} \frac{a_n}{b_n} = c
     with
    0<c<0 < c < \infty
    .
  • If the infinite series
    bn\sum b_n
     converges and
    0anbn0 \le a_n \le b_n
     for all sufficiently large
    nn
    (that is, for all
    n>Nn > N
    for some fixed value
    NN
    ), then the infinite series
    an\sum a_n
     also converges.
  • If the infinite series
    bn\sum b_n
     diverges and
    0anbn0 \le a_n \le b_n
     for all sufficiently large
    nn
    , then the infinite series
    an\sum a_n
     also diverges.


Key Terms

  • integral test: a method used to test infinite series of non-negative terms for convergence by comparing it to improper integrals
  • improper integral: an integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity


Comparison tests may mean either limit comparison tests or direct comparison tests. The limit comparison test is a method of testing for the convergence of an infinite series, while the direct comparison test is a way of deducing the convergence or divergence of an infinite series or an improper integral by comparison with other series or integral whose convergence properties are already known.

Limit Comparison Test

Statement: Suppose that we have two series,
Σnan\Sigma_n a_n
 and
Σnbn\Sigma_n b_n
, where
ana_n
,
bnb_n
 are greater than or equal to
00
for all
nn
. If
limnanbn=c\lim_{n \to \infty} \frac{a_n}{b_n} = c
 with
0<c<0 < c < \infty
, then either both series converge or both series diverge.

Example: We want to determine if the series
Σn+12n2\Sigma \frac{n+1}{2n^2}
 converges or diverges. For this we compare it with the series
Σ1n\Sigma \frac{1}{n}
, which diverges. As
limnn+12n2n1=12\lim_{n \to \infty} \frac{n+1}{2n^2} \frac{n}{1} = \frac{1}{2}
, we have that the original series also diverges.

image

Limit Convergence Test: The ratio between

n+12n2\frac{n+1}{2n^2}
and
1n\frac{1}{n}
for
nn \rightarrow ∞
is
12\frac{1}{2}
. Since the sum of the sequence
1n\frac{1}{n}
(i.e., 1n)\left ( \text{i.e., }\sum {\frac{1}{n}}\right)
diverges, the limit convergence test tells that the original series (with
n+12n2\frac{n+1}{2n^2}
) also diverges.

Direct Comparison Test

The direct comparison test provides a way of deducing the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known. In this atom, we will check the series case only.

For sequences
{an}\{a_n\}
,
{bn}\{b_n\}
 with non-negative terms:

  • If the infinite series
    bn\sum b_n
     converges and
    0anbn0 \le a_n \le b_n
     for all sufficiently large
    nn
     (that is, for all
    n>Nn>N
    for some fixed value
    NN
    ), then the infinite series
    an\sum a_n
     also converges.
  • If the infinite series
    bn\sum b_n
     diverges and
    anbn0a_n \ge b_n \ge 0
     for all sufficiently large
    nn
    , then the infinite series
    an\sum a_n
     also diverges.


Example

The series
Σ1n3+2n\Sigma \frac{1}{n^3 + 2n}
 converges because
1n3+2n<1n3\frac{1}{n^3 + 2n} < \frac{1}{n^3}
 for
n>0n > 0
and
Σ1n3\Sigma \frac{1}{n^3}
 converges.

Alternating Series

An alternating series is an infinite series of the form
n=0(1)nan\sum_{n=0}^\infty (-1)^n\,a_n
or
n=0(1)n1an\sum_{n=0}^\infty (-1)^{n-1}\,a_n
with
an>0a_n > 0
for all
nn
.

Learning Objectives

Describe the properties of an alternating series and the requirements for one to converge

Key Takeaways

Key Points

  • The theorem known as the "Leibniz Test," or the alternating series test, tells us that an alternating series will converge if the terms
    ana_n
    converge to
    00
    monotonically.
  • The signs of the general terms alternate between positive and negative.
  • The sum
    n=1(1)n+1n\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}
    converges by the alternating series test.


Key Terms

  • monotone: property of a function to be either always decreasing or always increasing
  • Cauchy sequence: a sequence whose elements become arbitrarily close to each other as the sequence progresses


An alternating series is an infinite series of the form:

n=0(1)nan\displaystyle{\sum_{n=0}^\infty (-1)^n\,a_n}


or:

n=0(1)n1an\displaystyle{\sum_{n=0}^\infty (-1)^{n-1}\,a_n}


with
an>0a_n > 0
for all
nn
. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.

Alternating Series Test

The theorem known as the "Leibniz Test," or the alternating series test, tells us that an alternating series will converge if the terms
ana_n
converge to
00
monotonically.

Proof: Suppose the sequence
ana_n
converges to
00
and is monotone decreasing. If
mm
is odd and
SmSn<amS_m - S_n < a_{m}
via the following calculation:

SmSn=k=0m(1)kakk=0n(1)kak =k=m+1n(1)kak=am+1am+2+am+3am+4++an=am+1(am+2am+3)anam+1am[an decreasing].\begin{align} S_m - S_n & = \sum_{k=0}^m(-1)^k\,a_k\,-\,\sum_{k=0}^n\,(-1)^k\,a_k\ \\& = \sum_{k=m+1}^n\,(-1)^k\,a_k \\ & =a_{m+1}-a_{m+2}+a_{m+3}-a_{m+4}+\cdots+a_n\\ & =\displaystyle a_{m+1}-(a_{m+2}-a_{m+3}) -\cdots-a_n \le a_{m+1}\le a_{m} \\& \quad [a_{n} \text{ decreasing}]. \end{align}


Since
ana_n
is monotonically decreasing, the terms are negative. Thus, we have the final inequality
SmSnamS_m - S_n \le a_{m}
. Similarly, it can be shown that, since
ama_m
converges to
00
,
SmSnS_m - S_n
converges to
00
for
m,nm, n \rightarrow \infty
. Therefore, our partial sum
SmS_m
converges. (The sequence
{Sm}\{ S_m \}
is said to form a Cauchy sequence, meaning that elements of the sequence become arbitrarily close to each other as the sequence progresses.) The argument for
mm
even is similar.

Example:

n=1(1)n+1n=112+1314+\displaystyle{\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots}


an=1na_n = \frac1n
converges to 0 monotonically. Therefore, the sum
n=1(1)n+1n\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}
converges by the alternating series test.

image

Alternating Harmonic Series: The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).

Absolute Convergence and Ratio and Root Tests

An infinite series of numbers is said to converge absolutely if the sum of the absolute value of the summand is finite.

Learning Objectives

State the conditions when an infinite series of numbers converge absolutely

Key Takeaways

Key Points

  • A real or complex series
    n=0an\textstyle\sum_{n=0}^\infty a_n
    is said to converge absolutely if
    n=0an=L\textstyle\sum_{n=0}^\infty \left|a_n\right| = L
    for some real number
    LL
    .
  • The root test is a convergence test of an infinite series that makes use of the limit
    L=limnan+1anL = \lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|
    .
  • The root test is a criterion for the convergence of an infinite series using the limit superior
    C=lim supnannC = \limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}
    .


Key Terms

  • summand: something which is added or summed
  • improper integral: an integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity
  • limit superior: the supremum of the set of accumulation points of a given sequence or set


An infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute value of the summand is finite. More precisely, a real or complex series
n=0an\textstyle\sum_{n=0}^\infty a_n
is said to converge absolutely if
n=0an=L\textstyle\sum_{n=0}^\infty \left|a_n\right| = L
for some real number
LL
. Similarly, an improper integral of a function,
0f(x)dx\textstyle\int_0^\infty f(x)\,dx
, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if
0f(x)dx=L\int_0^\infty \left|f(x)\right|dx = L
.

Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess, yet is broad enough to occur commonly. (A convergent series that is not absolutely convergent is called conditionally convergent.)

Ratio Test

The ratio test is a test (or "criterion") for the convergence of a series
n=1an\sum_{n=1}^\infty a_n
, where each term is a real or complex number and
ana_n
is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test.

The usual form of the test makes use of the limit,
L=limnan+1anL = \lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|
. The ratio test states that,

  • if
    L<1L < 1
    , then the series converges absolutely;
  • if
    L>1L > 1
    , then the series does not converge;
  • if
    L=1L = 1
    or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.


Root Test

The root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity
lim supnann\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}
, where
ana_n
are the terms of the series, and states that the series converges absolutely if this quantity is less than one but diverges if it is greater than one. It is particularly useful in connection with power series.

The root test was developed first by Augustin-Louis Cauchy and so is sometimes known as the Cauchy root test, or Cauchy's radical test. For a series
n=1an\sum_{n=1}^\infty a_n
, the root test uses the number
C=lim supnannC = \limsup_{n\rightarrow\infty}\sqrt[n]{ \left|a_n \right|}
, where "lim sup" denotes the limit superior, possibly ∞. Note that if
limnann\lim_{n\rightarrow\infty}\sqrt[n]{ \left|a_n \right|}
converges, then it equals
CC
and may be used in the root test instead. The root test states that

  • if
    C<1C < 1
    , then the series converges absolutely;
  • if
    C>1C > 1
    , then the series diverges;
  • if
    C=1C = 1
    and the limit approaches strictly from above, then the series diverges;
  • otherwise the test is inconclusive (the series may diverge, converge absolutely, or converge conditionally).


There are some series for which
C=1C = 1
and the series converges, e.g.:

1n2\displaystyle{\sum{\frac{1}{n^2}}}


and there are others for which
C=1C = 1
and the series diverges, e.g.:

1n\displaystyle{\sum{\frac{1}{n}}}


image

Ratio Test: In this example, the ratio of adjacent terms in the blue sequence converges to

L=12L=\frac{1}{2}
. We choose
r=L+12=34r = \frac{L+1}{2} = \frac{3}{4}
. Then the blue sequence is dominated by the red sequence for all
n2n \geq 2
. The red sequence converges, so the blue sequence does as well.

Tips for Testing Series

Convergence tests are methods of testing for the convergence or divergence of an infinite series.

Learning Objectives

Formulate three techniques that will help when testing the convergence of a series

Key Takeaways

Key Points

  • There is no single convergence test which works for all series out there.
  • Practice and training will help you choose the right test for a given series.
  • We have learned about the root /ratio test, integral test, and direct/ limit comparison test.


Key Terms

  • conditional convergence: A series or integral is said to be conditionally convergent if it converges but does not converge absolutely.


Convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series. When testing the convergence of a series, you should remember that there is no single convergence test which works for all series. It is up to you to guess and pick the right test for a given series. Practice and training will help you in expediting this "guessing" process.

Here is a summary for the convergence test that we have learned:

List of Tests

Limit of the Summand: If the limit of the summand is undefined or nonzero, then the series must diverge.

Ratio test: For
r=limnan+1anr = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|
, if
r<1r < 1
, the series converges; if
r>1r > 1
, the series diverges; if
r=1r = 1
, the test is inconclusive.

Root test: For
r=lim supnannr = \limsup_{n \to \infty}\sqrt[n]{ \left|a_n \right|}
, if
r<1r < 1
, then the series converges; if
r>1r > 1
, then the series diverges; if
r=1r = 1
, the root test is inconclusive.

Integral test: For a positive, monotone decreasing function
f(x)f(x)
such that
f(n)=anf(n)=a_n
, if
1f(x)dx=limt1tf(x)dx<\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty
then the series converges. But if the integral diverges, then the series does so as well.

image

Integral Test: The integral test applied to the harmonic series. Since the area under the curve

y=1xy = \frac{1}{x}
for
x[1,)x \in [1, ∞)
is infinite, the total area of the rectangles must be infinite as well.

Direct comparison test: If the series
n=1bn\sum_{n=1}^\infty b_n
is an absolutely convergent series and
anbn\left |a_n \right | \le \left | b_n \right|
for sufficiently large
nn
, then the series
n=1an\sum_{n=1}^\infty a_n
converges absolutely.

Limit comparison test: If
{an},{bn}>0\left \{ a_n \right \}, \left \{ b_n \right \} > 0
, and
limnanbn\lim_{n \to \infty} \frac{a_n}{b_n}
exists and is not zero, then
n=1an\sum_{n=1}^\infty a_n
converges if and only
ifn=1bnif \sum_{n=1}^\infty b_n
converges.

Power Series

A power series (in one variable) is an infinite series of the form
f(x)=n=0an(xc)nf(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n
, where
ana_n
is the coefficient of the
nn
th term and
xx
varies around
cc
.

Learning Objectives

Express a power series in a general form

Key Takeaways

Key Points

  • Power series usually arise as the Taylor series of some known function.
  • In many situations
    cc
    is equal to zero—for instance, when considering a Maclaurin series. In such cases, the power series takes the simpler form
    f(x)=n=0anxn=a0+a1x+a2x2+a3x3+f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots
    .
  • A power series will converge for some values of the variable
    xx
    and may diverge for others. If there exists a number
    rr
    with
    0<r0 < r \leq \infty
    such that the series converges when
    xc<r\left| x-c \right| <r
     and diverges when
    xc>r\left| x-c \right| >r
    , the number
    rr
    is called the radius of convergence of the power series.


Key Terms

  • Z-transform: transform that converts a discrete time-domain signal into a complex frequency-domain representation
  • combinatorics: a branch of mathematics that studies (usually finite) collections of objects that satisfy specified criteria and their structures


A power series (in one variable) is an infinite series of the form:

f(x)=n=0an(xc)n=a0+a1(xc)1+a2(xc)2+\displaystyle{f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c)^1 + a_2 (x-c)^2 + \cdots}


where
ana_n
 represents the coefficient of the
nn
th term,
cc
 is a constant, and
xx
 varies around
cc
 (for this reason one sometimes speaks of the series as being centered at
cc
). This series usually arises as the Taylor series of some known function. Any polynomial can be easily expressed as a power series around any center
cc
, albeit one with most coefficients equal to zero. For instance, the polynomial

f(x)=x2+2x+3f(x) = x^2 + 2x + 3


can be written as a power series around the center
c=1c=1
 as:

f(x)=6+4(x1)+1(x1)2+0(x1)3+0(x1)4+f(x) = 6 + 4 (x-1) + 1(x-1)^2 + 0(x-1)^3 + 0(x-1)^4 + \cdots \,


or, indeed, around any other center
cc
.

image

Exponential Function as a Power Series: The exponential function (in blue), and the sum of the first

n+1n+1
terms of its Maclaurin power series (in red).

In many situations
cc
is equal to zero—for instance, when considering a Maclaurin series. In such cases, the power series takes the simpler form

f(x)=n=0anxn=a0+a1x+a2x2+a3x3+\displaystyle{f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots}


These power series arise primarily in real and complex analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the
ZZ
-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument
xx
 fixed at
110\frac{1}{10}
. In number theory, the concept of
pp
-adic numbers is also closely related to that of a power series.

Radius of Convergence

A power series will converge for some values of the variable
xx
 and may diverge for others. All power series
f(x)f(x)
in powers of
(xc)(x-c)
will converge at
x=cx=c
. If
cc
is not the only convergent point, then there is always a number
rr
with 0 < r ≤ ∞ such that the series converges whenever
xc<r\left| x-c \right| <r
 and diverges whenever
xc>r\left| x-c \right| >r
. The number
rr
is called the radius of convergence of the power series. According to the Cauchy-Hadamard theorem, the radius
rr
can be computed from

r1=limnan+1an\displaystyle{r^{-1}=\lim_{n\to\infty}\left|{a_{n+1}\over a_n}\right|}


if this limit exists.

Expressing Functions as Power Functions

A power function is a function of the form
f(x)=cxrf(x) = cx^r
where
cc
and
rr
are constant real numbers.

Learning Objectives

Describe the relationship between the power functions and infinitely differentiable functions

Key Takeaways

Key Points

  • Since all infinitely differentiable functions can be represented in power series, any infinitely differentiable function can be represented as a sum of many power functions (of integer exponents ).
  • Therefore, an arbitrary function that is infinitely differentiable is expressed as an infinite sum of power functions (
    xnx^n
    ) of integer exponent:
    f(x)=n=0f(n)(0)n!xnf(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n! } \, x^{n}
    .
  • Functions of the form
    f(x)=x3f(x) = x^{3}
    ,
    f(x)=x1.2f(x) = x^{1.2}
    ,
    f(x)=x4f(x) = x^{-4}
    are all power functions.


Key Terms

  • differentiable: having a derivative, said of a function whose domain and co-domain are manifolds
  • power law: any of many mathematical relationships in which something is related to something else by an equation of the form
    f(x)=axkf(x) = a x^k


A power function is a function of the form
f(x)=cxrf(x) = cx^r
where
cc
and
rr
are constant real numbers. Polynomials are made of power functions. Since all infinitely differentiable functions can be represented in power series, any infinitely differentiable function can be represented as a sum of many power functions (of integer exponents). The domain of a power function can sometimes be all real numbers, but generally a non-negative value is used to avoid problems with simplifying. The domain of definition is determined by each individual case. Power functions are a special case of power law relationships, which appear throughout mathematics and statistics.

The Taylor series of a real or complex-valued function
f(x)f(x)
 that is infinitely differentiable in a neighborhood of a real or complex number
aa
is the power series:

n=0f(n)(a)n!(xa)n\displaystyle{\sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n! } \, (x-a)^{n}}


where
n!n!
denotes the factorial of
nn
 and
fnaf^na
denotes the
nn
th derivative of
ff
 evaluated at the point
x=ax=a
. Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial. If the Taylor series is centered at zero, then that series is also called a Maclaurin series:

f(x)=n=0f(n)(0)n!xn\displaystyle{f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n! } \, x^{n}}


Therefore, an arbitrary function that is infinitely differentiable is expressed as an infinite sum of power functions (
xnx^n
) of integer exponent.

image

sinx\sin x
in Taylor Approximations: Figure shows
sinx\sin x
and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. As more power functions with larger exponents are added, the Taylor polynomial approaches the correct function.

Examples

Functions of the form
f(x)=x3f(x) = x^3
,
f(x)=x1.2f(x) = x^{1.2}
,
f(x)=x4f(x) = x^{-4}
 are all power functions.

Taylor and Maclaurin Series

Taylor series represents a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point.

Learning Objectives

Identify a Maclaurin series as a special case of a Taylor series

Key Takeaways

Key Points

  • Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
  • A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function.
  • The Taylor series of a real or complex-valued function
    f(x)f(x)
     that is infinitely differentiable in a neighborhood of a real or complex number a is the power series
    f(x)=n=0f(n)(a)n!(xa)nf(x)=n=0f(n)(a)n!(xa)nf(x)=∑∞n=0f(n)(a)n!(x-a)nf(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n! } \, (x-a)^{n}
    . If
    a=0a = 0
    , the series is called a Maclaurin series.


Key Terms

  • differentiable: having a derivative, said of a function whose domain and co-domain are manifolds
  • analytic function: a real valued function which is uniquely defined through its derivatives at one point


A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.

It is common practice to approximate a function by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error in this approximation. Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial. The Taylor series of a function is the limit of that function's Taylor polynomials, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function.

image

Exponential Function as a Power Series: The exponential function (in blue) and the sum of the first

n+1n+1
terms of its Taylor series at
00
(in red) up to
n=8n=8
.

The Taylor series of a real or complex-valued function
f(x)f(x)
that is infinitely differentiable in a neighborhood of a real or complex number
aa
 is the power series:

f(x)=f(a)+f(a)1!(xa)+f(a)2!(xa)2+=_n=0f(n)(a)n!(xa)n\begin{align} f(x) &= f(a)+\frac {f'(a)}{1! } (x-a)+ \frac{f''(a)}{2! } (x-a)^2+ \cdots \\ &= \sum\text{\textunderscore}{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n! } \, (x-a)^{n} \end{align}


where
n!n!
denotes the factorial of
nn
and
fnaf^na
denotes the
nn
th derivative of
ff
 evaluated at the point
x=ax=a
. The derivative of order zero
ff
is defined to be
ff
itself, and
(xa)0(x - a)^0
 and
0!0!
are both defined to be
11
. In the case that
a=0a=0
, the series is also called a Maclaurin series.

Example 1

The Maclaurin series for
(1x)1(1 - x)^{-1}
 for
x<1\left| x \right| < 1
 is the geometric series: 
1+x+x2+x3+ ⁣1+x+x^2+x^3+\cdots\!


so the Taylor series for
x1x^{-1}
 at
a=1a=1
 is:

1(x1)+(x1)2(x1)3+1-(x-1)+(x-1)^2-(x-1)^3+\cdots


Example 2

The Taylor series for the exponential function
exe^x
 at
a=0a=0
is:

1+x11!+x22!+x33!+x44!+x55!+=1+x+x22+x36+x424+x5120+ ⁣=n=0xnn!\displaystyle{1 + \frac{x^1}{1! } + \frac{x^2}{2! } + \frac{x^3}{3! } + \frac{x^4}{4! } + \frac{x^5}{5! }+ \cdots \\= 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \frac{x^5}{120} + \cdots\! \\= \sum_{n=0}^\infty \frac{x^n}{n! }}


Applications of Taylor Series

Taylor series expansion can help approximating values of functions and evaluating definite integrals.

Learning Objectives

Describe applications of the Taylor series expansion

Key Takeaways

Key Points

  • The partial sums of the series, which is called the Taylor polynomials, can be used as approximations of the entire function.
  • Differentiation and integration of power series can be performed term by term, and hence could be easier than directly working with the original function.
  • The (truncated) series can be used to compute function values numerically. This is particularly useful in evaluating special mathematical functions (such as Bessel function).


Key Terms

  • definite integral: the integral of a function between an upper and lower limit
  • complex analysis: theory of functions of a complex variable; a branch of mathematical analysis that investigates functions of complex numbers
  • analytic function: a real valued function which is uniquely defined through its derivatives at one point


Uses of the Taylor series for analytic functions include:

1. The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function. These approximations are often good enough if sufficiently many terms are included. Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.

image

Taylor Polynomials: As more terms are added to the Taylor polynomial, it approaches the correct function. This image shows

sinx\sin x
 and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

2. Differentiation and integration of power series can be performed term by term and is hence particularly easy. Taylor series is especially useful in evaluating definite integrals. For example, to evaluate
01ex3dx\int_{0}^{1} e^{x^3} \, dx
, the Taylor series expansion of
et=n=01n!tne^t= \sum_{n=0}^{\infty} \frac{1}{n! } \, t^n
and the substitution of
t=x3t=x^3
 can be used. Since each term in the summation can be integrated separately, we can evaluate the definite integral as long as the sum converges.

3. An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis available.

4. The (truncated) series can be used to compute function values numerically. This is particularly useful in evaluating special mathematical functions (such as Bessel function).

5. Algebraic operations can be done readily on the power series representation; for instance the Euler's formula follows from Taylor series expansions for trigonometric and exponential functions.
eixe^{ix}
can be found from the Taylor expansion of
cos(x)\cos(x)
 and
sin(x)\sin(x)
:

cos(x)=1x22!+x44!x66!+\displaystyle{\cos(x) = 1-\frac{x^2}{2!}+\frac{x^4}{4! } -\frac{x^6}{6! }+ \cdots}


sin(x)=xx33!+x55!x77!+\displaystyle{\sin(x) = x - \frac{x^3}{3!}+\frac{x^5}{5! } -\frac{x^7}{7! } + \cdots}


and adding the two terms together yields:

cos(x)+isin(x)=(1x22!+x44!)+i(xx33!+x55!)=1+ix+(ix)22!+(ix)33!+(ix)44!+=eix\begin{align} cos(x)+i\,sin(x) & = (1-\frac{x^2}{2!}+\frac{x^4}{4! } - \cdots) + i (x - \frac{x^3}{3! } + \frac{x^5}{5! } - \cdots) \\ & = 1 + ix + \frac{(ix)^2}{2! } + \frac{(ix)^3}{3! }+ \frac{(ix)^4}{4! } + \cdots \\ & = e^{ix} \end{align}


This result is of fundamental importance in many fields of mathematics (for example, in complex analysis), physics and engineering.

Summing an Infinite Series

Infinite sequences and series can either converge or diverge.

Learning Objectives

Describe properties of the infinite series

Key Takeaways

Key Points

  • Infinite sequences and series continue indefinitely.
  • A series is said to converge when the sequence of partial sums has a finite limit.
  • A series is said to diverge when the limit is infinite or does not exist.


Key Terms

  • limit: a value to which a sequence or function converges
  • sequence: an ordered list of objects


A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely. Unlike finite summations, infinite series need tools from mathematical analysis, and specifically the notion of limits, to be fully understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, and finance.

For any infinite sequence of real or complex numbers, the associated series is defined as the ordered formal sum

n=0an=a0+a1+a2+\displaystyle{\sum_{n=0}^{\infty}a_n = a_0 + a_1 + a_2 + \cdots}


The sequence of partial sums
Sk{S_k}
 associated to a series
n=0an\sum_{n=0}^\infty a_n
 is defined for each k as the sum of the sequence
an{a_n}
 from
a0a_0
 to
aka_k
:

Sk=n=0kan=a0+a1++ak\displaystyle{S_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k}


Infinite sequences and series can either converge or diverge. A series is said to converge when the sequence of partial sums has a finite limit. By definition the series
n=0an\sum_{n=0}^\infty a_n
 converges to a limit
LL
if and only if the associated sequence of partial sums  converges to
LL
. This definition is usually written as:

L=n=0anL=limkSk\displaystyle{L = \sum_{n=0}^{\infty}a_n \Leftrightarrow L = \lim_{k \rightarrow \infty} S_k}


If the limit of is infinite or does not exist, the series is said to diverge.

image

Infinite Series: An infinite sequence of real numbers shown in blue dots. This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. It is, however, bounded.

An easy way that an infinite series can converge is if all the
ana_{n}
 are zero for sufficiently large
nn
s. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense. Working out the properties of the series that converge even if infinitely many terms are non-zero is, therefore, the essence of the study of series. In the following atoms, we will study how to tell whether a series converges or not and how to compute the sum of a series when such a value exists.

Convergence of Series with Positive Terms

For a sequence
{an}\{a_n\}
, where
ana_n
 is a non-negative real number for every
nn
, the sum
n=0an\sum_{n=0}^{\infty}a_n
 can either converge or diverge to
\infty
.

Learning Objectives

Identify convergence conditions for a sequence with positive terms

Key Takeaways

Key Points

  • Because the partial sum
    SkS_k
     of a series with non-negative terms can only increase as
    kk
    becomes larger, the limit of the partial sum can either converge or diverge to
    \infty
    .
  • A geometric sum
    1+12+14+18++12n+1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots+ \frac{1}{2^n}+\cdots
     converges to
    22
    , which can be understood visually.
  • The series
    n11n2\sum_{n \ge 1} \frac{1}{n^2}
     is convergent. This can be seen by comparing individual terms of the series with a sequence
    (bn=1n11n)\left( b_n = \frac{1}{n-1} - \frac{1}{n} \right)
    , which is know to converge.


Key Terms

  • converge: of a sequence, to have a limit
  • convergence test: methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series


For a sequence
{an}\{a_n\}
, where
ana_n
 is a non-negative real number for every
nn
, the sequence of partial sums

Sk=n=0kan=a0+a1++akS_k = \sum_{n=0}^{k}a_n = a_0 + a_1 + \cdots + a_k


is non-decreasing. Because the partial sum
SkS_k
 can only increase as
kk
becomes larger, the limit of the partial sum can either converge or diverge to
\infty
. Therefore, it follows that a series
n=0an\sum_{n=0}^{\infty} a_n
 with non-negative terms converges if and only if the sequence
SkS_k
 of partial sums is bounded.

Example 1

The series
n11n2\sum_{n \ge 1} \frac{1}{n^2}
 is convergent because of the inequality:

1n21n11n,(n2)\displaystyle{\frac1 {n^2} \le \frac{1}{n-1} - \frac{1}{n}, (n \ge 2)}


and because:

n2(1n11n)=(112)+(1213)+(1314)+=1\displaystyle{\sum_{n \ge 2} \left(\frac{1}{n-1} - \frac{1}{n} \right) =\left(1-\frac{1}{2}\right) + \left(\frac{1}{2}-\frac{1}{3}\right) + \left(\frac{1}{3}-\frac{1}{4}\right) + \cdots = 1}


Example 2

Would the series

S=1+12+14+18++12n+\displaystyle{S = 1 + \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots+ \frac{1}{2^n}+\cdots}


converge? It is possible to "visualize" its convergence on the real number line? We can imagine a line of length
22
, with successive segments marked off of lengths
11
,
12\frac{1}{2}
,
14\frac{1}{4}
, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off
12\frac{1}{2}
, we still have a piece of length
12\frac{1}{2}
unmarked, so we can certainly mark the next
14\frac{1}{4}
. This argument does not prove that the sum is equal to
22
(although it is), but it does prove that it is at most
22
. In other words, the series has an upper bound. Proving that the series is equal to
22
requires only elementary algebra, however. If the series is denoted
SS
, it can be seen that:

S2=1+12+14+18+2=12+14+18+116+\displaystyle{\frac{S}{2} \,= \frac{1+ \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots}{2} \\ \quad= \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+ \frac{1}{16} +\cdots}


Therefore:

SS2=1S=2\displaystyle{S-\frac{S}{2} = 1\\S = 2}


image

Geometric Sum: Visualization of the geometric sum in Example 2. The length of the line (

22
) can contain all the successive segments marked off of lengths
11
,
12\frac{1}{2}
,
14\frac{1}{4}
, etc.

For these specific examples, there are easy ways to check the convergence. However, it could be the case that there are no easy ways to check the convergence. For these general cases, we can experiment with several well-known convergence tests (such as ratio test, integral test, etc.). We will learn some of these tests in the following atoms.

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