Convergence Tests for Series
Test for Divergence
Geometric Series
∞

𝑛𝑛=1
∞
� 𝑎𝑎𝑟𝑟
𝑛𝑛=0
Integral Test

� 𝑎𝑎𝑛𝑛
𝑛𝑛−1
∞
� 𝑎𝑎𝑛𝑛 where 𝑐𝑐 ≥ 0 and 𝑎𝑎𝑛𝑛 = 𝑓𝑓(𝑛𝑛) for all 𝑛𝑛
𝑛𝑛=𝑐𝑐
p-series
∞
�
Comparison Test
𝑛𝑛=1
1
𝑛𝑛𝑝𝑝
� 𝑎𝑎𝑛𝑛 and � 𝑏𝑏𝑛𝑛 where 0 ≤ 𝑎𝑎𝑛𝑛 ≤ 𝑏𝑏𝑛𝑛 for all 𝑛𝑛
Limit Comparison Test
� 𝑎𝑎𝑛𝑛 and � 𝑏𝑏𝑛𝑛 where 𝑎𝑎𝑛𝑛 , 𝑏𝑏𝑛𝑛 > 0 and lim
𝑎𝑎𝑛𝑛
𝑛𝑛→∞ 𝑏𝑏𝑛𝑛
Alternating Series Test
∞
�(−1)𝑛𝑛−1 𝑏𝑏𝑛𝑛 where 𝑏𝑏𝑛𝑛 > 0
= 𝑐𝑐 > 0
� 𝑎𝑎𝑛𝑛
Ratio Test
Root Test
𝑛𝑛→∞
If lim 𝑎𝑎𝑛𝑛 = 0, then inconclusive
𝑛𝑛→∞
𝑎𝑎


If |𝑟𝑟| < 1, the series converges to
1−𝑟𝑟
If |𝑟𝑟| ≥ 1, then the series diverges


𝑓𝑓(𝑛𝑛) must be continuous, positive, and decreasing
∞
If ∫𝑐𝑐 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 converges, then the series converges
∞
If ∫𝑐𝑐 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 diverges, then the series diverges



If p > 1, then the series converges
If p ≤ 1, then the series diverges


If ∑ 𝑏𝑏𝑛𝑛 converges, then ∑ 𝑎𝑎𝑛𝑛 converges
If ∑ 𝑎𝑎𝑛𝑛 diverges, then ∑ 𝑏𝑏𝑛𝑛 diverges



If ∑ 𝑏𝑏𝑛𝑛 converges, then ∑ 𝑎𝑎𝑛𝑛 converges
If ∑ 𝑎𝑎𝑛𝑛 diverges, then ∑ 𝑏𝑏𝑛𝑛 diverges
To find 𝑏𝑏𝑛𝑛 consider only the terms of 𝑎𝑎𝑛𝑛 that have
the greatest effect on the magnitude

Converges if 0 < bn+1 < bn for all n and lim 𝑏𝑏𝑛𝑛 = 0


If ∑|𝑎𝑎𝑛𝑛 | converges, then ∑ 𝑎𝑎𝑛𝑛 converges
If the series of absolute values ∑|𝑎𝑎𝑛𝑛 | is
convergent, then the series is absolutely
𝑛𝑛=1
Absolute Value Test
If lim 𝑎𝑎𝑛𝑛 ≠ 0, then the series diverges

𝑛𝑛→∞
convergent
If the series is convergent but not absolutely
convergent, then the series is conditionally
convergent
|𝑎𝑎𝑛𝑛+1 |
� 𝑎𝑎𝑛𝑛 with lim
= 𝐿𝐿
𝑛𝑛→∞ |𝑎𝑎𝑛𝑛 |
𝑛𝑛
� 𝑎𝑎𝑛𝑛 with lim �𝑎𝑎𝑛𝑛 = 𝐿𝐿
𝑛𝑛→∞



If L < 1, then the series converges absolutely
If L > 1 or L is infinite, then the series diverges
If L = 1, then the test is inconclusive



If L < 1, then the series converges absolutely
If L > 1 or L is infinite, then the series diverges
If L = 1, then the test is inconclusive
Test for Divergence
Does lim 𝑎𝑎𝑛𝑛 = 0?
𝑛𝑛→∞
P-Series
Flowchart for Convergence Tests for Series
∑an Diverges
No
Yes
Does an = 1/np, n ≥ 1?
Yes
Is p > 1
No
No
Geometric Series
Does an = arn-1, n ≥ 1
Yes
Is |r| < 1?
Does an =(-1)nbn or
an = (-1)n-1bn , bn ≥ 0?
∑an Diverges
∞
�
Yes
𝑎𝑎𝑛𝑛 =
𝑛𝑛=1
No
No
Alternating Series Test
∑an Converges
Yes
𝑎𝑎
1 − 𝑟𝑟
∑an Diverges
Yes
Is bn+1 ≤ bn & lim bn = 0
𝑛𝑛→∞
Yes
∑an Converges
No
Try one of the following Tests:
Comparison Test
Pick {bn}. Does ∑ bn
converge?
Limit Comparison Test
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 {𝑏𝑏𝑛𝑛 }. 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷
𝑎𝑎𝑛𝑛
lim
= 𝑐𝑐 > 0
𝑛𝑛→∞ 𝑏𝑏𝑛𝑛
c finite & an , bn > 0?
Integral Test
Does an = f(n), f(x) is
continuous, positive &
decreasing on [a, ∞)?
Ratio Test
𝐼𝐼𝐼𝐼 lim |𝑎𝑎𝑛𝑛+1 /𝑎𝑎𝑛𝑛 | ≠ 1?
𝑛𝑛→∞
Root Test
𝑛𝑛
𝐼𝐼𝐼𝐼 lim �|𝑎𝑎𝑛𝑛 | ≠ 1?
𝑛𝑛→∞
Yes
No
Yes
Is 0 ≤ an ≤ bn
Is 0 ≤ bn ≤ an
∞
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 � 𝑏𝑏𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐?
𝑛𝑛=1
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 � 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐?
Yes
𝐼𝐼𝐼𝐼 lim |𝑎𝑎𝑛𝑛+1 /𝑎𝑎𝑛𝑛 | < 1?
Yes
𝑎𝑎
𝑛𝑛→∞
𝑛𝑛
𝐼𝐼𝐼𝐼 lim �|𝑎𝑎𝑛𝑛 | < 1?
𝑛𝑛→∞
∑an Diverges
Yes
∑an Converges
Yes
No
Yes
∞
Yes
∑an Converges
Yes
No
Yes
No
Yes
No
∑an Diverges
∞
�
𝑛𝑛=𝑎𝑎
𝑎𝑎𝑛𝑛 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
∑an Diverges
∑an Abs. Conv.
∑an Diverges
∑an Abs. Conv.
∑an Diverges