Notes on Differentiating and Integrating Power Series
Since a power series can be considered as a function o its IOC, it is natural to ask: How does one perform certain operations on them that are cus
Notes on the Comparison Tests
There are two comparison tests. One is called The Comparison Test and the other one is called The Limit Comparison Test. Both tests require one to choose another series t
Notes on the Differentiation of the Natural Logarithm Function
This lesson is entirely about differentiating function involving the natural logarithm function. The homework problems make use of the ru
Notes on the Generalized Exponential and Logarithmic Functions
It turns out that all exponential and logarithmic functions are related to the Natural Exponential and Natural Logarithm functions. Mathe
Notes on the Graph of the Natural Logarithm Function
dt called the natural logarithm function. Notice the t 1 domain of this function is the interval ( 0, ) since the function is continuous on the t 1
Notes on the Integral Test
The Integral Test is a method for determining the convergence or divergence of a series of positive terms. It has the advantage that, unlike some of the other tests we will
Notes on the Interval of Convergence for a Power Series
Definition: Let cn ( x a ) denote a power series about x = a . The Interval of
n n =0
Convergence (sometimes denoted by IOC) for cn ( x a ) is
Notes on the Natural Exponential Function
From your experience in high school logarithms are intimately connected to exponentials. This lesson shows how the natural logarithm leads to the idea of the
Notes on the Radius of Convergence for a Power Series
Definition: A power series is a series of the form cn ( x a ) , where a is a constant,
n n =0
to as a power series centered about a (or a power s
Notes on the Ratio Test The Ratio Test provides a method to verify a given series converges absolutely. Unlike most of our previous tests, it applies to any series. Since absolute convergence implies
Steps for Integration by Partial Fractions p( x) dx , where p and q are polynomials Problem Statement: Evaluate q( x)
n n 1 with no common factors. So p ( x ) = an x + an 1 x + L + a0 , an 0 , and q (
Notes on the Calculus of the Natural Exponential Function
This lesson concerns the differentiation and integration of functions involving the natural exponential function.
x Differentiation: Consider
Notes on the Alternating Series Test
The Alternating Series Test applies only to a series with a very special form. Namely, the series must be of the form an , where an an +1 < 0 . That is, the terms
Further Notes on Integrating Tangents and Secants
In many applications, particularly those involving arclength and surface area, one 2n 2 k +1 xdx , where n and k are needs to evaluate integrals of th
Further Notes on Sequences
Additional Methods of Determining Convergence/Divergence of a Sequence: Using rules to evaluate limits: Just as there are rules for evaluating limits of functions, there are
Integration by Parts
Integration by Parts is the name of a technique for integrating certain types if functions. It is based on the Product Rule for differentiation, namely ( uv ) = uv + uv . Integrat
Notes on Absolute Convergence
So far the only series that convergence tests apply are either series of positive terms or alternating series. There are series that do not fit either category. The notio
Notes on Integration involving the Natural Logarithm Function
As you learned in calculus I, differentiation formulas read backwards yield d integration formulas, recall for example sin x = cos x cos x
Notes on Sequences
Definition: A sequence is an ordered list of numbers: a, b, c, d , K . Each number is called a term of the sequence and is referred to by the number of its position in the listing.
Notes on Series - Continued
Convergent series may be thought of as finite sums as they sum to a finite number. Therefore we can have an arithmetic associated with them. Divergent series though do not
Notes on Series
Definitions: Let cfw_ an n =1 be a sequence. Define a new sequence cfw_ sn n =1 , called the sequence
of partial sums for the sequence cfw_ an n =1 , by the following equations:
s
Notes on Taylor Polynomials - Continued
Definition: Suppose f is a function which has derivatives up to order N at a point a. Then n f ( k) ( a) k for each 0 n N , define the polynomial Tn ( x ) ( x a
Taylor Series for a Function
Up until now we have studied power series as defining a function on the interval of convergence. One may turn the issue around though and ask the following question: Given