Harold’s Series Convergence Tests
“Cheat Sheet”
5 August 2015
1
2
Divergence or nth Term Test
3
Geometric Series Test
Series: ∑∞
𝑛=1 𝑎𝑛
𝑛
Series: ∑∞
𝑛=0 𝑎𝑟
Condition(s) of Convergence:
None. This test cannot be
used to show convergence.
Condition of Convergence:
|𝑟| < 1
Condition(s) of Divergence:
lim 𝑎𝑛 ≠ 0
* Sum: 𝐒 = lim
𝑎(1−𝑟 𝑛 )
𝑛→∞
4
Series: ∑∞
𝑛=1
Condition of Divergence:
|𝑟| ≥ 1
𝑛→∞
1−𝑟
=
𝑛+1
Series: ∑∞
𝑎𝑛
𝑛=1 (−1)
Condition of Convergence:
0 < 𝑎𝑛+1 ≤ 𝑎𝑛
lim 𝑎𝑛 = 0
𝑛→∞
or if ∑∞
𝑛=0 |𝑎𝑛 | is convergent
Condition of Divergence:
None. This test cannot be
used to show divergence.
* Remainder: |𝑅𝑛 | ≤ 𝑎𝑛+1
7
Root Test
Series: ∑∞
𝑛=1 𝑎𝑛
Condition of Convergence:
𝑛
lim √|𝑎𝑛 | < 1
𝑛→∞
Condition of Divergence:
𝑛
lim √|𝑎𝑛 | > 1
𝑛→∞
* Test inconclusive if
𝑛
lim √|𝑎𝑛 | = 1
𝑛→∞
Series: ∑∞
𝑛=1 (𝑎𝑛+1 − 𝑎𝑛 )
Condition of Convergence:
lim 𝑎𝑛 = 𝐿
𝑛→∞
Condition of Divergence: None
Condition of Convergence:
𝑝>1
𝑎
1−𝑟
6
Integral Test
Series: ∑∞
𝑛=1 𝑎𝑛
when 𝑎𝑛 = 𝑓(𝑛) ≥ 0
and 𝑓(𝑛) is continuous, positive and
decreasing
Condition of Convergence:
∞
∫1 𝑓(𝑥)𝑑𝑥 converges
Condition of Divergence:
∞
∫1 𝑓(𝑥)𝑑𝑥 diverges
* Remainder: 0 < 𝑅𝑁 ≤
8
∞
∫𝑁 𝑓(𝑥)𝑑𝑥
Ratio Test
Series: ∑∞
𝑛=1 𝑎𝑛
Condition of Convergence:
𝑎𝑛+1
lim |
|<1
𝑛→∞ 𝑎𝑛
Condition of Divergence:
𝑎𝑛+1
lim |
|>1
𝑛→∞ 𝑎𝑛
* Test inconclusive if
𝑎𝑛+1
lim |
|=1
𝑛→∞ 𝑎𝑛
9
Direct Comparison Test
Limit Comparison Test
(𝑎𝑛 , 𝑏𝑛 > 0)
({𝑎𝑛 }, {𝑏𝑛 } > 0)
Series: ∑∞
𝑛=1 𝑎𝑛
Series: ∑∞
𝑛=1 𝑎𝑛
Condition of Convergence:
0 < 𝑎𝑛 ≤ 𝑏𝑛
and ∑∞
𝑛=0 𝑏𝑛 is absolutely
convergent
Condition of Convergence:
𝑎𝑛
lim
=𝐿>0
𝑛→∞ 𝑏𝑛
and ∑∞
𝑛=0 𝑏𝑛 converges
Condition of Divergence:
0 < 𝑏𝑛 ≤ 𝑎𝑛
and ∑∞
𝑛=0 𝑏𝑛 diverges
Condition of Divergence:
𝑎𝑛
lim
=𝐿>0
𝑛→∞ 𝑏𝑛
∞
and ∑𝑛=0 𝑏𝑛 diverges
NOTE: These tests prove
convergence and divergence, not
the actual limit L or sum S.
10
Telescoping Series Test
1
𝑛𝑝
Condition of Divergence:
𝑝≤1
5
Alternating Series Test
p - Series Test
NOTE:
1) May need to reformat with partial
fraction expansion or log rules.
2) Expand first 5 terms. n=1,2,3,4,5.
3) Cancel duplicates.
4) Determine limit L by taking the
limit as 𝑛 → ∞.
5) Sum: 𝑆 = 𝑎1 − 𝐿
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
1
Sequence: lim 𝑎𝑛 = 𝐿
𝑛→∞
(𝑎𝑛 , 𝑎𝑛+1 , 𝑎𝑛+2 , …)
Series: ∑∞
𝑛=1 𝑎𝑛 = 𝐒
(𝑎𝑛 + 𝑎𝑛+1 + 𝑎𝑛+2 + ⋯ )
Choosing a Convergence Test for Infinite Series
Courtesy David J. Manuel
Do
the individual
terms approach 0?
No
Series Diverges by
the Divergence Test.
Yes
Does
the series
alternate
signs?
Do individual
terms have
factorials or
exponentials?
Yes
No
No
Is individual term
easy to integrate?
Use Integral Test
Yes
No
Do individual terms
involve fractions with
powers of n?
No
Yes
Use Alternating
Series Test
(Do absolute value of
terms go to 0?)
Use Comparison Test
or Limit Comp. Test
(Look at dominating
terms)
Copyright © 2011-2015 by Harold Toomey, WyzAnt Tutor
2
Use Ratio Test
Yes
(Ratio of Consecutive
Terms)