A Study Guide on Convergence Tests for Series of Non-negative Terms
Produced by Andrew Boneff and Bobby Crismyre
Revised by Learning Center
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The following gives some guidelines for checking the convergence of a series  an for an  0 .
n 1
1
Check to see if the series is geometric or of the form  p . If not, then perform one of the
n 1 n
following tests.
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Tests
Divergence Test
Let lim an  L
n 
Guidelines and Hints
Should be first test you try, if limit is
hard move on
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If L  0 ,  an diverges
n2  3
diverges
2
n 1 3n  n

 1
Use geometric series or  p for  bn
n 0
n 0 n
Typically use facts like: lnn  n ,
cos2 n  1and sin2 n  1
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Example: 
n 0
If L  0 , test fails
Direct Comparison Test


If  bn converges and an  bn then  an converges
n 0
n 0


n 0
n 0
If  bn diverges and an  bn then  an diverges

Example: 
n 1
Limit Comparison Test
a
Let lim n  c
n 
bn

1
for  bn
p
n 0 n
n 0
Works well on polynomials, power of
polynomials, fractions powers of n.

Use geometric series or 


n 0
n 0
If c  0 , then  an and  bn both converge or both diverge
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
If c  0 and  bn converges, then  an converges
n 0
n 0


n 0
n 0

Example: 
n 1
If c   and  bn diverges, then  an diverges
Ratio Test
a
Let lim n 1  
n 
an
n2  3
n5  n
converges
Works well on factorials and
exponentials
Will not work on polynomials, powers
of polynomials
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If   1 ,  an converges
n 0
2n  5
Example:  n converges
n 0
3


If   1,  an diverges
n 0
If   1, test fails
Integral Test
Let an  f (n)
Check that f (n) is positive, continuous and decreasing on N ,  


1
n 1
If so, then  f  x  dx and  an either both converge or both
diverge
5
diverges
5n  1
Use last since conditions are a pain to
check. Difficult to use since must be
able to integrate function
1
diverges
n 1 n ln n
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Example: 