Name__________________________________
Period ________
Calculus
Lesson 9.1 Infinite Series Day 1

Goal: To determine if the series
a
n 1
Is lim an  0 ?
Nth Term Test
n
converges.
NO
n 
Series diverges
YES

Geometric Series
Test
Is
a
n 1
n
 a  ar  ar 2  ...
YES
Series converges to
a
1 r
NO
Find a pattern for S n and evaluate lim S n .
n 
Rules for Convergent and Divergent Series
If the sequence of partial sums  Sn  converges to a limit L, we say the series CONVERGES and that its sum is
L. The lim Sn  L .
n 
If the sequence of partial sums does not converge, we say the series DIVERGES.
nth term test

a
n
n1
DIVERGES if lim an fails to exist or is different from zero.
n 
However, you cannot assume that if lim an does exist or is zero then the series converges.
n 
Infinite Sum for Geometric Series
If an is geometric (of the form an  a1r n 1 ) and if r  1 then the series converges to
S
a1
1 r
Determine if the infinite series converges. If it does converge, state the value.


1
1. 
2.  n 2
n 1 n( n  1)
n 1
4.
9
9
9
9



 ...
2
3
100 100 100 100 4

5.
n 1

6.
1

n 1 n
 5
  2

(Harmonic Series)
7.

n 1
n

1

3n 
1
1

n
n 1

3.
n
 n 1
n 1
Homework: p483: 2 ab, 7-13 odd, 18-20
Determine if the following series converge, and if so, to what value.


6
6
1. 
2. 
n 1  n  1 n  2 
n  0  2n  1 2n  1

3.
 n 
 ln  n  1 
n 1