Series Convergence/Divergence Flow Chart
TEST FOR DIVERGENCE
Does limn→∞ an= 0? PanDiverges
NO
p-SERIES
Does an= 1/np,n≥1?
YES
Is p > 1?
YES PanConverges
YES
PanDiverges
NO
GEOMETRIC SERIES
Does an=arn−1,n≥1?
NO
Is |r|<1?
YES P∞
n=1 an=a
1−r
YES
PanDiverges
NO
ALTERNATING SERIES
Does an= (−1)nbnor
an= (−1)n−1bn,bn≥0?
NO
Is bn+1 ≤bn& lim
n→∞
bn= 0?
YES PanConverges
YES
TELESCOPING SERIES
Do subsequent terms cancel out previous terms in the
sum? May have to use partial fractions, properties
of logarithms, etc. to put into appropriate form.
NO
Does
lim
n→∞
sn=s
sfinite?
YES Pan=s
YES
PanDiverges
NO
TAYLOR SERIES
Does an=f(n)(a)
n!(x−a)n?
NO
Is xin interval of convergence?
YES P∞
n=0 an=f(x)
YES
PanDiverges
NO
Try one or more of the following tests:
NO
COMPARISON TEST
Pick {bn}. Does Pbnconverge?
Is 0 ≤an≤bn?
YES PanConverges
YES
Is 0 ≤bn≤an?
NO
NO
PanDiverges
YES
LIMIT COMPARISON TEST
Pick {bn}. Does lim
n→∞
an
bn=c > 0
cfinite & an, bn>0? Does
∞
X
n=1
bnconverge?
YES PanConverges
YES
PanDiverges
NO
INTEGRAL TEST
Does an=f(n), f(x) is contin-
uous, positive & decreasing on
[a, ∞)?
Does Z∞
a
f(x)dx converge?
YES P∞
n=aanConverges
YES
PanDiverges
NO
RATIO TEST
Is limn→∞ |an+1/an| 6= 1? Is lim
n→∞
an+1
an
<1?
YES PanAbs. Conv.
YES
PanDiverges
NO
ROOT TEST
Is limn→∞
n
p|an|6= 1? Is lim
n→∞
n
p|an|<1?
YES PanAbs. Conv.
YES
PanDiverges
NO
Series and Sequences Cheat Sheet
