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 produ
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
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
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 o
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
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 comp
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 l
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
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
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 divergen
(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
c
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
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
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
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 vect
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 t