Flowchart summarizing most series convergence tests (designed by E. Kim) in alternate colors
Look at the
sequence of
terms {an }
Start P
with
series
an
sequence diverges
not zero
sequence converges
What is
lim an ?
zero
n→∞
no
Is an =
1
np ?
yes
no
no
P
Series
an diverges
by nth term test
P
Is
an a geometric series?
yes
Let r be the
common ratio.
1
np is a
p-series.
P
Is every
an ≥ 0?
P
|r| < 1
P
an diverges by
geometric
series test
|r| ≥ 1
P
p>1
an converges
by the p-test
p≤1
yes
Consider a function f (x) where
an = f (n), i.e., replace Reach n with
∞
an x. Can you integrate N f (x) dx?
yes
R ∞Does
f (x) dx
N
converge
or diverge?
an converges by
geometric series test
conv.
P
an diverges
by the p-test
P
an converges
by integral test
div.
no, I don’t know how
P
Is there a sequence
{cn } where an ≤ cn
for all n > N ?
Does
P the series
cn converge
or diverge?
yes
div.
no
Is there a sequence
{dn } where an ≥ dn
for all n > N ?
Does
P the series
dn converge
or diverge?
yes
conv.
no
an diverges
by integral test
conv.
P
an converges by
comparison test
div.
seq. conv.
an+1
n→∞ an
ρ = lim
n→∞
P
Is
an an alternating series?
an diverges
by the ratio test
ρ<1
P
an converges
by the root test
ρ>1
an diverges
by the root test
yes
P
P
Huh?
no
no
yes
no
Let un = |an |. Is un ≥
un+1 for all n ≥ N ?
Does un → 0?
an
ρ=1
seq. div.
yes
√
n
P
ρ>1
sequence diverges
seq. conv.
an converges
by the ratio test
ρ = lim
ρ=1
Consider the se√
quence { n an }
an diverges by
comparison test
P
ρ<1
Considernthe seo
quence aan+1
n
P
DoesPthe
series
|an |
converge?
an converges
by the absolute
convergence test
no
yes
P
an converges
by the alternating series test