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