Flowchart summarizing most series convergence test (designed by E. Kim)
Look at the
sequence of
terms {an }
Start P
with
series
an
not zero
sequence converges
What is
lim an ?
zero
n→∞
no
Is an =
1
np ?
yes
no
no
P
Series
an
diverges by the
Test for Divergence
sequence diverges
P
Is
an a geometric series?
yes
Let r be the
common ratio.
P
Is every
an ≥ 0?
1
np
is a
p-series.
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
{bn } where an ≤ bn
for all n > N ?
yes
div.
no
Is there a sequence
{bn } where an ≥ bn
for all n > N ?
yes
conv.
no
Consider
o
nthe se
quence aan+1
n
Does
P the series
bn converge
or diverge?
Does
P the series
bn converge
or diverge?
Consider p
the sequence { n |an |}
an+1 seq. conv.
L = lim n→∞
an seq. conv.
L = lim
|an |
L=1
P
Is
an an alternating series?
P
an converges by
comparison test
div.
P
an diverges by
comparison test
L<1
an converges
by the ratio test
P
L>1
an diverges
by the ratio test
L<1
P
L>1
an converges
by the root test
P
an diverges
by the root test
Huh?
no
no
yes
no
Let bn = |an |. Is bn ≥
bn+1 for all n ≥ N ?
Does bn → 0?
p
n
n→∞
seq. div.
yes
conv.
P
L=1
sequence diverges
an diverges
by integral test
DoesPthe
series
|an |
converge?
yes
P
an converges
by the absolute
convergence test
no
yes
Notation and terminology follow J. Stewart,
Calculus: Early Transcendentals (7th ed.) as closely as possible
P
an converges
by the alternating series test