1
Series: ∑ 𝑎𝑛
∞
𝑛=1
None. This test cannot be used to
show convergence.
Condition(s) of Divergence:
lim
𝑛→∞ 𝑎𝑛≠ 0
2
Series: ∑ 𝑎𝑟𝑛∞
𝑛=0
Condition of Convergence:
|𝑟|< 1
Sum: 𝐒 = lim
𝑛→∞ 𝑎(1−𝑟𝑛)
1−𝑟 = 𝑎
1−𝑟
Condition of Divergence:
|𝑟|≥ 1
3
Series: ∑ 1
𝑛𝑝
∞
𝑛=1
Condition of Convergence:
𝑝 > 1
Condition of Divergence:
𝑝 ≤ 1
4
Series: ∑ (−1)𝑛+1 𝑎𝑛
∞
𝑛=1
Condition of Convergence:
0 < 𝑎𝑛+1 ≤ 𝑎𝑛
lim
𝑛→∞ 𝑎𝑛= 0
or if ∑ |𝑎𝑛|
∞
𝑛=0 is convergent
Condition of Divergence:
None. This test cannot be used
to show divergence.
* Remainder: |𝑅𝑛|≤ 𝑎𝑛+1
5
Series: ∑ 𝑎𝑛
∞
𝑛=1
when 𝑎𝑛= 𝑓(𝑛)≥ 0
and 𝑓(𝑛) is continuous, positive and
decreasing
Condition of Convergence:
∫𝑓(𝑥)𝑑𝑥
∞
1 converges
Condition of Divergence:
∫𝑓(𝑥)𝑑𝑥
∞
1 diverges
* Remainder: 0 < 𝑅𝑁≤∫𝑓(𝑥)𝑑𝑥
∞
𝑁
6
Series: ∑ 𝑎𝑛
∞
𝑛=1
Condition of Convergence:
lim
𝑛→∞ |𝑎𝑛+1
𝑎𝑛| < 1
Condition of Divergence:
lim
𝑛→∞ |𝑎𝑛+1
𝑎𝑛| > 1
* Test inconclusive if
lim
𝑛→∞ |𝑎𝑛+1
𝑎𝑛| = 1
7
Series: ∑ 𝑎𝑛
∞
𝑛=1
Condition of Convergence:
lim
𝑛→∞ √|𝑎𝑛|
𝑛< 1
Condition of Divergence:
lim
𝑛→∞ √|𝑎𝑛|
𝑛> 1
* Test inconclusive if
lim
𝑛→∞ √|𝑎𝑛|
𝑛= 1
8
(𝑎𝑛,𝑏𝑛> 0)
Series: ∑ 𝑎𝑛
∞
𝑛=1
Condition of Convergence:
0 < 𝑎𝑛 ≤ 𝑏𝑛
and ∑ 𝑏𝑛
∞
𝑛=0 is absolutely
convergent
Condition of Divergence:
0 < 𝑏𝑛 ≤ 𝑎𝑛
and ∑ 𝑏𝑛
∞
𝑛=0 diverges
9
({𝑎𝑛},{𝑏𝑛} > 0)
Series: ∑ 𝑎𝑛
∞
𝑛=1
Condition of Convergence:
lim
𝑛→∞ 𝑎𝑛
𝑏𝑛= 𝐿 > 0
and ∑ 𝑏𝑛
∞
𝑛=0 converges
Condition of Divergence:
lim
𝑛→∞ 𝑎𝑛
𝑏𝑛= 𝐿 > 0
and ∑ 𝑏𝑛
∞
𝑛=0 diverges
10
Series: ∑ (𝑎𝑛+1 −𝑎𝑛)
∞
𝑛=1
Condition of Convergence:
lim
𝑛→∞ 𝑎𝑛= 𝐿
Condition of Divergence: None
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
NOTE: These tests prove
convergence and divergence, not
the actual limit 𝐿 or sum S.
Sequence: lim
𝑛→∞ 𝑎𝑛= 𝐿
(𝑎𝑛,𝑎𝑛+1, 𝑎𝑛+2,…)
Series: ∑ 𝑎𝑛
∞
𝑛=1 = 𝐒
(𝑎𝑛+𝑎𝑛+1 + 𝑎𝑛+2 +⋯)
Condition(s) of Convergence:
Geometric Series Test p - Series Test
Alternating Series Test
Integral Test Ratio Test
Root Test
Direct Comparison Test Limit Comparison Test
Telescoping Series Test
Divergence or nth Term Test
cheat sheet
