Series Convergence/Divergence Flow Chart

Series Convergence/Divergence Flow Chart
TEST FOR DIVERGENCE
Does limn→∞ an = 0?
p-SERIES
Does an = 1/np YES
, n ≥ 1?
NO
GEOMETRIC SERIES
Does an = ar n−1 , n ≥ 1?
NO
ALTERNATING SERIES
Does an = (−1) n bn or
an = (−1) n−1 bn, bn ≥ 0?
NO
YES
YES
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
TAYLOR SERIES
Does an = f(n) (a)
n! (x − a) n ?
NO
Try one or more of the following tests:
COMPARISON TEST
Pick {bn}. Does bn converge?
LIMIT COMPARISON TEST
Pick {bn}. Does lim
n→∞
c finite & an, bn > 0?
an
bn
= c > 0
INTEGRAL TEST
Does an = f(n), f(x) is continuous,
positive & decreasing on
[a, ∞)?
RATIO TEST
YES
YES
NO
NO
YES
YES
NO
Is p > 1?
Is |r| < 1?
Is bn+1 ≤ bn & lim
n→∞ bn = 0?
YES
Does
lim
n→∞ sn = s
s finite?
Is x in interval of convergence?
Does
Does
Is 0 ≤ an ≤ bn?
Is 0 ≤ bn ≤ an?
∞
∞
bn converge?
n=1
Is limn→∞ |an+1/an| = 1? Is lim
n→∞
an+1
YES
an
ROOT TEST
Is limn→∞ n |an| = 1?
YES
a
f(x)dx converge?
< 1?
n Is lim |an| < 1?
n→∞
YES
NO
YES
NO
YES
YES
YES
YES
NO
YES
NO
YES
NO
YES
NO
YES
NO
YES
NO
an Diverges
an Converges
an Diverges
∞
n=1 an = a
1−r
an Diverges
an Converges
an = s
an Diverges
∞
n=0 an = f(x)
an Diverges
an Converges
an Diverges
an Converges
an Diverges
∞
n=a an Converges
an Diverges
an Abs. Conv.
an Diverges
an Abs. Conv.
an Diverges

Problems 1-38 from Stewart’s Calculus, page 784
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
∞
n=1
∞
n=1
∞
n=1
n 2 − 1
n 2 + n
n − 1
n 2 + n
1
n 2 + n
∞
n−1 n − 1
(−1)
n2 + n
n=1
∞ (−3) n+1
n=1
∞
n=1
∞
n=2
∞
k=1
∞
k=1
∞
n=1
∞
n=2
2 3n
n 3n
1 + 8n
1
n ln(n)
2 k k!
(k + 2)!
k 2 e −k
n 2 e −n3
(−1) n+1
n ln(n)
∞
(−1) n n
n2 + 25
n=1
∞ 3nn2 n=1
n!
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
∞
sin(n)
n=1
∞
n=0
∞
n=1
∞
n=1
∞
n=2
n!
2 · 5 · 8 · · · · · (3n + 2)
n 2 + 1
n 3 + 1
(−1) n 2 1/n
(−1) n−1
√ n − 1
∞
n ln(n)
(−1) √
n
n=1
∞
k=1
k + 5
5 k
∞ (−2) 2n
n=1
∞
n=1
n n
√ n 2 − 1
n 3 + 2n 2 + 5
∞
tan(1/n)
n=1
∞
n=1
∞
e
n=1
n2
∞
n=1
cos(n/2)
n 2 + 4n
n!
n 2 + 1
5 n
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
∞
k=1
∞
n=1
∞
n=1
k ln(k)
(k + 1) 3
e 1/n
n 2
tan −1 (n)
n √ n
∞
(−1) j
√
j
j + 5
j=1
∞
k=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=2
5 k
3 k + 4 k
(2n) n
n 2n
sin(1/n)
√ n
1
n + n cos 2 (n)
2
n
n
n + 1
1
(ln(n)) ln(n)
∞
( n√ 2 − 1) n
n=1
∞
( n√ 2 − 1)
n=1