Tests for Series: A Summary
Test
Series
Interpretation
Divergence
Test
If lim an  0
a
Geometric
Series
 ar
n
n 

n 0
p-series

Integral Test
Diverges if
p

1
,

p
n 1 n
r 1
These are useful
comparison series for the
Basic Comparison and
Limit Comparison Tests.
r 1


n 1
 log n 
Converges or diverges with
q
np
,

 f  x  dx
decreasing
If
Series that closely
resemble p-series or
geometric series
Series that resemble
p-series or geometric
series (like BCT)
Useful when it is
difficult to show that
an  bn or an  bn but
you are pretty sure the
behavior is similar.
Ratio Test
Series involving
n
n
n !, n , a or
expressions like
1 3  5  ...   2n 1
f  n   an for all n
f  x  continuous and
1
an  bn and bn converges, then
an also converges.
If
an  bn and bn diverges, then
an also diverges.
Limit
Comparison
Test (LCT)
Cannot ever be used to
determine convergence,
only divergence
Diverges if p  1
or similar series
Basic
Comparison
Test (BCT)1
n 
Converges if p  1
1
n
n 1
diverges if lim an  0
Converges if
n
Comments
an
 L then
n  b
n
If lim
n
n
L  0 and bn converges then
= series of which you
are trying to determine
convergence.
b
If L  0 then both series converge
or both series diverge.
If
a
= series of your
choice (usually p-series or
geometric), for which you
already know convergence
or divergence.
an also converges
an 1
 L then
n  a
n
If lim
If L  1 then series converges
absolutely
Test fails if L  1 . Need to
use another test. If series
has terms alternating in
sign, try the Alternating
Series Test.
If L  1 or infinite then series
diverges
Alternating
Series Test
  1 a ,
 cos  n  an
n
n
If terms are alternating in sign,

lim an  0 then the series
n
converges
1
2

decreasing an 1  an , and
Called the Comparison Test in Stewart
Recall that absolute convergence implies convergence.
Useful if the series
converges but does NOT
converge absolutely. Other
tests may be easier if
series converges
absolutely2.