Summary of Convergence and Divergence Tests for Series
Test
nth -term
Series
Convergence or Divergence
Comments
�
Diverges if lim an �= 0
Inconclusive if lim an = 0
an
Geometric
Series
∞
�
arn−1
∞
�
1
np
n=1
Integral
∞
�
an
n=1
an = f (n)
�
Comparison
n→∞
(i) Converges with sum S =
n=1
p-series
n→∞
an ,
�
bn
an > 0, bn > 0
(ii) Diverges if |r| ≥ 1
a
if |r| < 1
1−r
Ratio
Root
Alternating
Series
�
|an |
�
an
nth term an of a series is similar
to arn−1
Useful for comparison tests if the
(i) Converges if p > 1
(ii) Diverges if p ≤ 1
nth term an of a series is similar
to
�∞
(i) Converges if 1 f (x)dx converges
�∞
(ii) Diverges if 1 f (x)dx diverges
(i) if
�
1
np
The function f obtained from
an = f (n) must be continuous,
positive, decreasing, integrable.
bn converges and an ≤ bn for
�
every n, then
an converges.
�
(ii) If
bn diverges
and an ≥ bn for
�
every n, then
an diverges.
(iii) If limn→∞ (an /bn ) = c for some
positive real number c, then
both series converge or both diverge
The comparison series
an+1
| = L, the series
n→∞ an
(i) converges (absolutely) if L < 1
(ii) diverges is L > 1(or ∞)
Inconclusive if L = 1.
If lim |
�
Useful for comparison tests if the
If lim
n→∞
�
n
|an | = L, (or ∞), the series
�
bn is
often a geometric series or a
p-series. To find bn in (iii)
consider only the terms of an that
have the greatest effect on the
magnitude.
Useful of an involves factorials
or nth powers.
If an > 0 for every n, disregard
the absolute value.
Inconclusive if L = 1
an
(i)converges (absolutely) if L < 1
(ii) diverges is L > 1(or ∞)
Useful if an involves nth powers
If an > 0 for every n, disregard
the absolute value.
(−1)n an
an > 0
Converges if ak ≥ ak+1 for every k
and lim an = 0
Applicable only to an
alternating series.
�
�
an
n→∞
If
�
|an | converges, then
�
an converges.
Useful for series that contain
both positive and negative terms.