1. Convergence and Divergence Tests for Series
Test
When to Use
Conclusions
∞
Divergence Test
for any series
Diverges if lim |an | =
6 0.
an
n→∞
n=0
∞
Integral Test
X
X
∞
Z
an with an ≥ 0 and an decreasing
f (x)dx and
1
n=0
∞
X
an both converge/diverge
n=0
where f (n) = an .
Comparison Test
∞
X
an and
n=0
∞
X
∞
X
bn
n=0
∞
n=0
X
if 0 ≤ an ≤ bn
Limiting Comparison Test
∞
X
an , (an > 0). Choose
n=0
n→∞
if lim
n→∞
if lim
n→∞
Convergent test
∞
X
∞
X
an
=L
bn
∞
X
an converges.
n=0
∞
an diverges =⇒
n=0
X
bn diverges.
n=0
bn , (bn > 0)
n=0
if lim
bn converges =⇒
with 0 < L < ∞
∞
X
an and
n=0
∞
an
=0
bn
X
an
=∞
bn
X
∞
X
bn both converge/diverge
n=0
bn converges =⇒
n=0
∞
an converges.
n=0
∞
bn diverges =⇒
n=0
(−1)n an (an > 0)
∞
X
X
an diverges.
n=0
converges if
n=0
for alternating Series
lim an = 0 and an is decreasing
n→∞
∞
∞
Absolute Convergence
for any series
X
an
If
n=0
X
|an | converges, then
n=0
∞
X
an converges,
n=0
(definition of absolutely convergent series.)
Conditional Convergence
for any series
∞
X
an
if
n=0
∞
X
∞
X
Calculate lim
n→∞
Root Test:
Calculate lim
n→∞
an converges.
an conditionally converges
n=0
an ,
there are 3 cases:
n=0
Ratio Test:
∞
X
n=0
n=0
∞
X
For any series
|an | diverges but
an+1
=L
an
if L < 1, then
p
n
if L > 1, then
|an | = L
∞
X
|an | converges ;
n=0
∞
X
|an | diverges;
n=0
if L = 1, no conclusion can be made.
2. Important Series to Remember
Series
How do they look
Conclusions
∞
Geometric Series
X
arn
n=0
p-series
∞
X
1
n=1
np
Converges to
a
if |r| < 1 and diverges if |r| ≥ 1
1−r
Converges if p > 1 and diverges if p ≤ 1