Summary of Convergence and Divergence Tests for Series
Test
Divergence
1
P
Series
Convergence or Divergence
Diverges if lim an 6= 0
an
Geometric
1
P
p-series
ar
Integral
(2) Diverges if |r| > 1
an
if the nth -term an of a series
is similar to arn .
if the nth -term an of a series
(2) Diverges if p 6 1
is similar to
1
P
bn converges and an 6 bn
for every n, then
1
P
bn
n=1
an > 0, bn > 0
(2) If
1
P
n=1
1
P
1
.
np
The function f obtained
(1) Converges if 1 f (x) dx converges;
R1
(2) Diverges if 1 f (x) dx diverges
n=1
n=1
if |r| < 1;
(1) Converges if p > 1;
(1) If
an ,
r
R1
an = f (n)
1
P
1
Useful for comparison tests
n=1
Comparison
a
(1) Converges to S =
1 1
P
p
n=1 n
1
P
n!1
Useful for comparison tests
n
n=0
series
Inconclusive if lim an = 0
n!1
n=1
Comments
an converges;
n=1
bn diverges and an > bn
1
P
for every n, then
an diverges;
n=1
an
(3) If lim
> 0 (not 1), then
n!1 bn
both series converge or both diverge.
from an = f (n) must be
continuous, positive,
decreasing, and readily
integrable.
The comparison series
1
P
bn is often a geometric
n=1
series or a p-series. To find
bn in (3), consider only
the terms of an that have
the greatest e↵ect on the
magnitude.
Inconclusive if L = 1. Useful
Ratio
1
P
an
n=1
an+1
If lim
= L (or 1), the series
n!1
an
if an involves factorials or
(1) converges (absolutely) if L < 1;
nth powers. If an > 0 for
(2) diverges if L > 1 (or 1)
every n, the absolute value
sign may be disregarded.
If lim
Root
1
P
n!1
an
n=1
Alternating
1
P
( 1)n an
n=1
Series
an > 0
p
n
|an | = L (or 1), the series
Inconclusive if L = 1. Useful
if an involves nth powers.
(1) converges (absolutely) if L < 1;
If an > 0 for every n, the
(2) diverges if L > 1 (or 1)
absolute value sign may be
disregarded.
Converges if ak > ak+1 for every k
Applicable only to an
and lim an = 0.
alternating series.
n!1