Convergence or Divergence of
Infinite Series
1
Definition of Convergence for an infinite series:

Let
a
n 1
n
be an infinite series of positive terms.
The series converges if and only if the sequence of partial
sums,

S n  a1  a2  a3  a , converges. This means: lim S n   a n
n 
n 1
Divergence Test:

If lim a  0 , the series
n
n 
a
n 1

Example: The series 
n 1
However,

n
n 1
2
n
diverges.
is divergent since lim
n 
n
n 1
2
 lim
n 
1
1 1
1
n2
lim a n  0 does not imply convergence!
n 
2
Special Series
3
Geometric Series:
The Geometric Series:
a  ar  ar 2    ar n 1   converges for 1  r  1
If the series converges, the sum of the series is:

7
Example: The series  5 
n 1  8 
n
with a  a1 
35
8
a
1 r
and r 
7
8
converges .
The sum of the series is 35.
4
p-Series:

The Series:
1

p
n
n 1
and diverges for
(called a p-series) converges for
p 1
p 1
Example: The series

n
n 1
1
1.001

is convergent. The series  1
n 1 n
is divergent.
5
Convergence Tests
6
11.3
The Integral Test
11.4
The Comparison Tests
11.5
Alternating Series
11.6
Ratio and Root Tests
Integral test:
If f is a continuous, positive, decreasing function on
[1, ) with f (n)  a n

then the series
converges.
a
n 1

n
converges if and only if the improper integral
 f ( x)dx
1

Example: Try the series:
1

3
n 1 n
Note: in general for a series of the form:
8
Comparison test:

If the series a n
b
is convergent and
b
is divergent and
n 1

(b)If
n 1
•
•
and
n 1

(a)If

n
n
b
n 1
are two series with positive terms, then:
n

a n  bn
an  bn
for all n, then
for all n, then
a
n 1

a
n 1
n
n
converges.
diverges.
(smaller than convergent is convergent)
(bigger than divergent is divergent)



3n
3n
1


3



2
2
n2 n  2
n2 n
n2 n
Examples:
which is a divergent harmonic series. Since the
original series is larger by comparison, it is
divergent.


5n
5n 5  1
 3   2

3
2
2 n 1 n
2
n

n

1
n 1
n 1 2n
which is a convergent p-series. Since the
original series is smaller by comparison, it is
convergent.
9
Limit Comparison test:


If the the series
a
n 1
where
0c
and
n
b
n 1
n
an
c
n  b
n
are two series with positive terms, and if lim
then either both series converge or both series diverge.
Useful trick: To obtain a series for comparison, omit lower order terms in the numerator and the
denominator and then simplify.

n
Examples: For the series  2
compare to
n 1 n  n  3

For the series
n  n
3
n 1
n
 n2
compare to
n
n 1
n
2


n 1
1
n
3
2
which is a convergent p-series.
   which is a divergent geometric series.

 


n
n 1 3
n 1  3 

n


n
10
Alternating Series test:

If the alternating series
n 1



1
bn

n 1
bn  bn1
and
 b1  b2  b3  b4  b5  b6  
satisfies:
lim bn  0 then the series converges.
n 
Definition: Absolute convergence means that the series converges without alternating
(all signs and terms are positive).

Example:
The series

n 0
 1n
n 1
is convergent but not absolutely convergent.
Alternating p-series
(1) n converges for p > 0.

p
n 1 n
Example: The series
convergent.
(1) n and the Alternating Harmonic series

n
n 1


(1) n

n
n 1

are
11
Ratio test:
a
(a)If lim n1  1 then the series
n a
n
an1
1
n a
n
(b)If lim

a
n 1
n
converges;
the series diverges.
(c)Otherwise, you must use a different test for convergence.
If this limit is 1, the test is inconclusive and a different test is required.
Specifically, the Ratio Test does not work for p-series.
Example:
12
11.7
Strategy for Testing Series
13
Summary:
Apply the following steps when testing for convergence:
1. Does the nth term approach zero as n approaches infinity? If not, the
Divergence Test implies the series diverges.
2. Is the series one of the special types - geometric, telescoping, pseries, alternating series?
3. Can the integral test, ratio test, or root test be applied?
4. Can the series be compared in a useful way to one of the special
types?
14
Summary
of all
tests:
15
Example
Determine whether the series
diverges using the Integral Test.
converges or
Solution: Integral Test:
Since this improper integral is divergent, the series
 (ln n)/n is also divergent by the Integral Test.
16
More examples
17
Example 1
Since as an  ≠ as n 
Divergence.
, we should use the Test for
18
Example 2: Using the limit comparison test
Since an is an algebraic function of n, we compare the
given series with a p-series.
The comparison series for the Limit Comparison Test is
where
19
Example 3
Since the series is alternating, we use the Alternating
Series Test.
20
Example 4
Since the series involves k!, we use the Ratio Test.
21
Example 5
Since the series is closely related to the geometric series
, we use the Comparison Test.
22