Math 152 Section 10.2 Series
For a given sequence a n n  1 ,


a n or
 a n is the sequence, s n  where
n 1
n
sn 
 a k is the sum of the first n terms of the given sequence, { a n } .
k 1
s n is called the nth partial sum.

Example:

1
n 1 n
s1  1,
s2  1 
is the notation for the sequence of partial sums s n 
2
1
4
,
s3  1 
1
4

1
9
,
s4  1 
1

4
1
9

1
, ...
16

If
If
lim s n
n 
lim s n
n 
exists and is equal to S, the series converges to S and
 a n =S.
n 1
does not exist, the series diverges.

Example 1: Show that the harmonic series
1
 n diverges.
n 1
Example 2:

The Geometric Series,
r
n
 1 r  r
2
3
 r  ...
converges if and only if | r | < 1.
n0

If | r |<1 then
r 
n0
n
1
1 r
.

Example 3: Find the sum of the series
2
 
5
n  0 


Example 4: Find the sum of the series

n0
53
4

n
n

n
.
2
 
5
n 1  

n 1
Some series are collapsing, also called telescoping.

Example:
1
 n ( n  1) Consider the general term,
n 1
1
n ( n  1)
n
sn 

1
n

1
1
 k ( k  1) = 1  2
k 1
which collapses into
n ( n  1)
.
We can write;
n 1
1
1
1

1

2
1
n 1
1

3
1

3
1
 ... 
4
1

n
1
n 1
which converges to 1. This means the series converges and
its sum is 1.

Example 5: Find the sum of the series

n 1 n
1
2
 2n
.

Theorem: If
 a n converges then
n 1
The Divergence Test: If
lim a n  0
n 
lim a n  0 .
n 
then the series
 an
diverges.
Examples:
n
 n2
diverges since the general term converges to 1, not 0.
n
 (  1) diverges since the general term does not converge.
Linear Property of Convergent Series:

If

n 1

an  A
and
 bn
n 1
 B
then
 c  a n  d  b n   cA  dB .
Example 6: Find the sum of the series

5 2  2 3
n 1
4

n
n
n
.