Series Convergence Flowchart
X
Things to remember:
an
• p-series: The series
n
does an → 0?
X 1
np
n
Diverges
by Divergence Test
converges if and only if p > 1
• Geometric series: If |r| < 1 then
Is an > 0?
∞
X
Is it alternating in sign
and |an | decreasing?
n=0
• All that matters is what happens on a tail.
For example, if an is only decreasing after
N then you can write:
∞
X
Try to show
P absolute convergence
i.e
|an | converges
an =
n=1
N
X
an +
n=1
Try Limit Comparison Test:
an
lim
= c then:
bn
P
P
if 0 < c < ∞ then
an converges ⇔ bn converges
P
P
if c = 0 and
bn converges then
an converges.
P
P
if c = ∞ and
bn diverges then
an diverges.
p
Is n |an |
easy to analyze?
Try Root Test:
p
n
lim |an | = c
P
if 0 ≤ c < 1 then
an converges
P
if c > 1 then
an diverges
if c = 1 then test is inconclusive
an+1 Is an easy to analyze?
Try
Ratio Test:
an+1 =c
lim an P
if 0 ≤ c < 1 then
an converges
P
if c > 1 then
an diverges
if c = 1 then test is inconclusive
P
Try Integral Test:
R∞
f (x) dx converges
an converges ⇔
∞
X
an
N +1
the finite sum clearly converges, and then
you can use convergence tests that require
decreasing on the other part.
Try Comparison Test:
P
P
if an < bn and
bn converges then
an converges
P
P
if bn < an and
bn diverges then
an diverges
Does it feel like an
‘looks like’
some bn ?
Is there a positive,
decreasing f (x)
where Rf (n) = an
and f (x) dx
isn’t so bad?
a
1−r
otherwise, that series diverges.
Converges
by Alternating Series Test
Are there any
easy comparisons?
arn =
Key
Yes
No