Flow Chart for
nth term test
for
divergence
lim a n = 0 ?
∑a
n
Convergence
∑a
NO
n→∞
n
diverges
YES
Geometric
Series?
Is
∞
∑ ar
∑ an =
∞
= ∑ ar ?
n −1
n =1
∞
n
n =0
Yes
If -1 < r < 1 then
n −1
=
n =1
∞
(look for n as an exponent…)
∑ ar
Otherwise
∑ ar
n −1
a
1− r
diverges
n =1
NO
Is
P-Series?
Or is
∑a
∞
n
∑ an =
=
1
∑n
n =1
∞
p
∑a
1
∑ n(ln n)
n =1
∞
?
p
? ( p>0)
Yes
n =1
∞
n
∑a
n =1
converges if p > 1
n
diverges if p ≤ 1
NO
Telescoping
Series?
If you were asked to find ∫ a n ,
would you use partial fractions?
Yes
This could be a telescoping series,
try writing out a sample of the
series using its partial fractions
and see if terms cancel out.
NO
Use
tests for
positive
series?
Is an > 0 ( or an < 0) for all n?
Go to page 3, TOP
NO
(mixed-sign series)
YES
Ratio
test?
a
Does lim n+1 = ρ
n →∞ a
n
Where 0 < ρ < ∞, ρ≠1?
∞
∑a
Yes
n =1
∞
∑a
n =1
NO
Go to
page 2
p. 1
n
converges if ρ < 1
n
diverges if ρ > 1
From p. 1,
still
assuming
an > 0
( or an < 0)
for all n
Limit Comparison Test:
a
Does lim bn = c ,
n→ ∞
Comparison
test(s)
Does
∑b
n
∑a
n
for 0 < c < ∞ or c=∞ ?
n
closely resemble
= p-series or geometric series?
Yes
If 0 < c < ∞, then ∑an and ∑bn
either both diverge or both converge.
If c = 0 and ∑bn converges,
then ∑an converges.
If c = ∞ and ∑bn diverges,
then ∑an diverges.
NO
LUCK?
Direct Comparison Test:
Does an < bn for all n > N ?
Then if ∑bn converges, so does ∑an.
or
Does an > bn for all n > N.
Then if ∑bn diverges, so does ∑an.
STILL
NO LUCK?
Integral
Test
Can you find f (x) such that f (n) = an?
Is f (x) always continuous, positive, and
decreasing for some N ≥ n
Then
Yes
∞
∞
n= N
N
∑ an and
∫ f ( x)dx either
both diverge or both converge
p. 2
From p. 1
( assuming mixed sign series)
∑a
n
Absolute
Convergence
Does
∑a
n
converge?
[ Use positive series tests
(from p. 1) on ∑ a n . ]
Yes
Then
∑a
n
converges
NO
Alternating
Series
Test
Is
∑a
Alternating series test:
n
an alternating series?
Yes
NO
Yeah, well, good luck with that!
p. 3
If |an+1 | ≤ |an | and an → 0 ,
then ∑ a n converges