Series Convergence/Divergence Flow Chart
TEST FOR DIVERGENCE
Does limn→∞ an = 0?
P
NO
YES
p-SERIES
Does an = 1/np , n ≥ 1?
YES
NO
Is p > 1?
YES
P
GEOMETRIC SERIES
Does an = arn−1 , n ≥ 1?
n=1
P
NO
ALTERNATING SERIES
Does an = (−1)n bn or
an = (−1)n−1 bn , bn ≥ 0?
YES
Is bn+1 ≤ bn & lim bn = 0?
n→∞
an Diverges
P∞
YES
NO
Is |r| < 1?
YES
an Converges
P
NO
an Diverges
YES
P
an =
a
1−r
an Diverges
an Converges
NO
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.
YES
Does
lim sn = s
n→∞
s finite?
YES
NO
NO
TAYLOR SERIES
Does an =
f (n) (a)
n! (x
n
− a) ?
YES
YES
Is x in interval of convergence?
NO
NO
P
P
an Diverges
P∞
n=0
P
an = s
an = f (x)
an Diverges
Try one or more of the following tests:
COMPARISON TEST
P
Pick {bn }. Does
bn converge?
YES
Is 0 ≤ an ≤ bn ?
YES
Is 0 ≤ bn ≤ an ?
YES
NO
NO
LIMIT COMPARISON TEST
lim an
n→∞ bn
Pick {bn }. Does
c finite & an , bn > 0?
=c>0
INTEGRAL TEST
Does an = f (n), f (x) is continuous, positive & decreasing on
[a, ∞)?
RATIO TEST
Is limn→∞ |an+1 /an | =
6 1?
ROOT TEST
p
Is limn→∞ n |an | 6= 1?
YES
Does
∞
X
bn converge?
n=1
YES
Does
Z
∞
f (x)dx converge?
a
YES
YES
Is
lim an+1
n→∞ an
< 1?
p
Is lim n |an | < 1?
n→∞
YES
NO
YES
NO
YES
NO
YES
NO
P
an Converges
P
P
an Diverges
an Converges
P
P∞
an Diverges
n=a
P
P
an Diverges
an Abs. Conv.
P
P
an Converges
an Diverges
an Abs. Conv.
P
an Diverges
Problems 1-38 from Stewart’s Calculus, page 784
∞
X
n2 − 1
n2 + n
n=1
14.
2.
∞
X
n−1
2+n
n
n=1
15.
3.
∞
X
1
2
n +n
n=1
16.
∞
X
n2 + 1
n3 + 1
n=1
4.
∞
X
17.
∞
X
1.
(−1)n−1
n=1
5.
6.
7.
8.
18.
n
∞ X
3n
1 + 8n
n=1
19.
∞
X
20.
1
p
n=2 n ln(n)
∞
X
∞
X
10.
(−1)n 21/n
∞
X
(−1)n−1
√
n−1
n=2
∞
X
ln(n)
(−1)n √
n
n=1
∞
X
k+5
23.
∞
X
(−2)2n
nn
n=1
∞
X
29.
tan(1/n)
∞
X
tan−1 (n)
√
n n
n=1
√
j
(−1)
30.
j+5
j=1
∞
X
31.
∞
X
k=1
j
3k
5k
+ 4k
32.
∞
X
(2n)n
n2n
n=1
33.
∞
X
sin(1/n)
√
n
n=1
34.
∞
X
5k
k 2 e−k
1
n + n cos2 (n)
n=1
2
n
∞ X
n
35.
n+1
n=1
n=1
∞
X
(−1)n+1
11.
n ln(n)
n=2
n
12.
(−1) 2
n + 25
n=1
13.
∞
X
e1/n
28.
n2
n=1
n!
2
·
5
·
8
·
·
·
· · (3n + 2)
n=0
k=1
n=1
∞
X
∞
X
√
∞
X
n2 − 1
22.
n3 + 2n2 + 5
n=1
3
∞
X
k ln(k)
(k + 1)3
k=1
21.
n2 e−n
27.
n=1
2k k!
(k + 2)!
k=1
∞
X
sin(n)
n=1
∞
X
(−3)n+1
23n
n=1
k=1
9.
n−1
n2 + n
∞
X
n
∞
X
3 n n2
n!
n=1
∞
X
cos(n/2)
24.
n2 + 4n
n=1
∞
X
n!
25.
en2
n=1
26.
∞
X
n2 + 1
5n
n=1
∞
X
36.
1
(ln(n))ln(n)
n=2
37.
∞
X
√
n
( 2 − 1)n
n=1
38.
∞
X
√
n
( 2 − 1)
n=1