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Preview – This file contains a few of the convergence tests that are in the full
file.
Summary Version 1
Before doing anything else, ask, “Are the terms approaching 0?”
n
th
Term Test
If terms approach 0 If terms don’t approach 0
If signs alternate: If signs don’t alternate:
Series definitely
diverges
Probably converges.
Use Alternating Series
Test.
Don’t know.
Use other methods.
If it’s a geometric series
then look at the common ratio, r.
Geometric Series Test
If | r | < 1 If | r | ≥ 1
converges diverges
=== page 2 preview ===
If all terms are positive
( or if we are trying to figure out whether the original series converges absolutely —
but remember, even if it doesn’t converge absolutely, it might converge conditionally!)
then…
Ratio Test
(This one is usually helpful with #
n
and n! )
Find the limit, L , of ratio of “next” to “now” ( a n + 1 / a n
If L < 1
If L > 1
(including L = ∞)
If L = 1
converges diverges
Don’t know.
Try something
else.
Preview – This file contains a few of the convergence tests that are in
the full file.
Summary Version 2
Note: version 2 has a blank version for students, as well as an answer key.
The preview below is the answer key; red text is absent in blank student version.
Name _____________________________
Period _____ Date
________________
Before you do anything else:
Test If… then…
n
th
term test
Consider what the
terms
approach as n gets
huge
i.e., find n
n
lim a
The terms approach 0
Don’t know. Keep
going!!!
The terms don’t approach 0 Diverges
Is it a special type of series?
Test If… then…
Geometric Series
test
Geometric series form:
0
n
n
a r
Look at common ratio,
r
(r is constant)
| r | < 1
Converges
initial
Sum
1 r
| r | > 1 Diverges
| r | = 1 ( r = 1 or -1)
5 + 5 + 5 + … or 5 + -5 + -5 + …
Diverges
unless it’s 0 + 0 + 0 + …
=== page 2 preview ===
Am I looking at a power series? Is there a factorial? Have all other ideas failed?
Test If… then…
Ratio Test
Find
n+
n
n
a
lim
a
Limit < 1 Converges [absolutely]
Limit > 1 Diverges
Limit = 1 Test fails. Try another
