Cauchys Inequality
Math 122 Calculus III
D Joyce, Fall 2012
Dot products in n-space. The dot product is an operation : Rn Rn R that takes
two vectors v and w and gives a scalar v w by adding the products of corresponding elements,
that is,
v w = (v1 , v2
Survey of Series and Sequences
Math 122 Calculus III
D Joyce, Fall 2012
The goal. One purpose of our study of series and sequences is to understand power series.
A power series is like a polynomial of innite degree. For example,
1 + x + x2 + + xn +
is a
Vectors
Math 122 Calculus III
D Joyce, Fall 2012
Vectors in the plane R2 . A vector v can be interpreted as an arrow in the plane R2 with
a certain length and a certain direction. The same vector can be moved around in the plane
if you dont change its len
Power Series
Math 122 Calculus III
D Joyce, Fall 2012
Introduction to power series. One of the main purposes of our study of series is to
understand power series. A power series is like a polynomial of innite degree. For example,
xn = 1 + x + x2 + + xn +
Series Convergence Tests
Math 122 Calculus III
D Joyce, Fall 2012
Some series converge, some diverge.
Geometric series. Weve already looked at these. We know when a geometric series
arn converges when its ratio r lies
converges and what it converges to. A
Summary: polar and parametric
Math 122 Calculus III
D Joyce, Fall 2012
This is a summary sheet about the topics weve discussed in polar coordinates and parametric equations.
Polar coordinates and complex numbers A point in the plane is described in rectan
LHpitals rule for 0 . The limit of this indetero
0
minant form depends on the rates that the numerator and denominator approach 0. If the numerator approaches 0 faster than the denominator, then
the limit will be small; if slower, large. Rates are
derivat
lim an = L, or more briey an L. Symbolically,
n
convergence says
> 0, N, n N, |an L| < .
Summary of denitions and theorems for A sequence that doesnt converge is said to diverge.
sequences
Note that limits of sequences are called discrete
Math 122 Calcul
Natural numbers
Math 122 Calculus III
D Joyce, Fall 2012
Well have occasion to distinguish between dierent kinds of numbers. Well consider the
natural numbers N, the integers Z, the rational numbers Q, the real numbers R, and the
complex numbers C.
The na
Analysis of the harmonic series and Eulers constant
Math 122 Calculus III
D Joyce, Fall 2012
Consider the standard harmonic series
n=1
1
1 1
1
= 1 + + + + +
n
2 3
n
1. We briey discussed the divergence of this series in class using Oresmes argument where
(cn ) converges, and so does
cn . Since
an
is the sum of two convergent series, it, too, converges.
q.e.d.
Alternating Series and Absolute
Convergence
Math 122 Calculus III
For example, the series
1
converges, so the
n2
(1)n
absolutely converges.
n2
sin n
Example 3 (a closed interval). S = [4, 9]. lub S =
9, glb S = 4. Like in the previous example, the lub
and the glb are the largest and smallest numbers in
the set. Any time S contains a largest number, that
number is its lub. Likewise, any time S contains
Determinants
Math 122 Calculus III
D Joyce, Fall 2012
What they are. A determinant is a value associated to a square array of numbers, that
square array being called a square matrix. For example, here are determinants of a general
2 2 matrix and a general
Exponentiating, we nd that
1
e n+1 1 +
e as the limit of (1 + 1/n)n
Math 122 Calculus III
1
n
1
en .
Taking the (n + 1)st power of the left inequality
gives us
1 n+1
e 1+ n
D Joyce, Fall 2012
th
This is a small note to show that the number e while taking
Cross products
Math 122 Calculus III
D Joyce, Fall 2012
The denition of cross products. The cross product : R3 R3 R3 is an operation
that takes two vectors u and v in space and determines another vector u v in space. (Cross
products are sometimes called o
More on Vectors
Math 122 Calculus III
D Joyce, Fall 2012
Unit vectors. A unit vector is a vector whose length is 1. If a unit vector u in the plane
R2 is placed in standard position with its tail at the origin, then its head will land on the
unit circle x