Review Outline for Math 121 Test 4
11.1 Introduction to Infinite Sequences
Be able to write out the first few terms of a sequence, given an nth term formula.
Given the terms of a sequence, be able to produce an nth term formula.
Sketch the graph of a sequence.
Give definition of the following terms and determine if an infinite sequence
a) is bounded above/below
d) has a limit
b) is increasing/decreasing
e) is geometric
c) is monotone
f) is arithmetic
Know how to use L’Hôpital’s rule, and the squeeze theorem to determine limits.
11.2 Introduction to Infinite Series
Know the difference between a sequence and a series.
Know the definition of convergence and divergence of series.
Understand and use the Divergence Test (and it’s converse) correctly.
Write terms of the partial sum sequence; use sigma notation to represent infinite series
Sum geometric sequences, when possible, and know when it is possible.
Know what the Harmonic Series is
Be able to sum telescoping series.
Be able to determine if an infinite series converges or diverges.
11.3, 11.4, 11.5, 11.6 Convergence tests for Infinite series
(See summary on next page)
11.7
Power Series
Use the ratio test to determine the radius of convergence for a
Be able to give the interval of convergence for a series
Given a power series for f(x), be able to write the series for f (g(x) ) (ie, if you know the series for sin x,
be able to produce the series for sin 2x or sin x2.)
11.8 Taylor and Maclaurin Series
Be able to find the Taylor or Maclaurin series of a given function
Memorize the Maclaurin series for sin x, cos x, and ex. (Also, geometric series with r = x)
Be able to multiply and divide power series; also to calculate limits by using infinite series
Summary of Series
Known Series

 ar
n1
(aka Geometric Series) converges to S 
n1

1
n
n1
p
a
, as long as -1 < r < 1
1 r

Nth term test
1
n
(aka p-series) converges only for p > 1

(aka harmonic series) diverges
n1
If
lim an  0, then the series diverges
If
lim an =0, then the series could converge or diverge
n
n


Integral Test Let f(n) = an. If
an > 0

f (x)dx converges/diverges, then
1
If
If
a
a
a
n
n1


a
If
a
If
diverges
n
n1

< known convergent series, then
n
converges
n1
n

> known convergent series, then no conclusion
n

< known divergent series, then no conclusion
n1



 an > known divergent series, then
n1


an > 0
converges/diverges
n
n1

Direct Comparison Test

a
n1

Limit Comparison Test

an > 0
n 
known
n 
If
known
n
Let lim
n 
an 1
an
conv
div

a
n
converges
n
diverges
n1
= L, L ≠ 0, L =  then

1) bn+1 < bn and
2) lim
Ratio Test
= L, L = 0, L ≠  then
an
lim
If
Alternating Series Test
an
lim
If

bn = 0,
then

 (1)
n1

a
n1
bn converges
n1

L
If L > 1, then
a
n
diverges
n
converges

n1

If L < 1, then
a
n1
If L =
1 , then the test fails—must use a different test
Growth hierarchy of functions
ln n  n
pos fraction  1
 n
pos whole number


exponentia
l
n
(c )

factorial
n!

variable base exponential
nn