Series Strategies and Tests for Convergence
Test Name
When to Use
Known Series – Geometric
When you recognize that the terms
are a constant raised to some power
Known Series – p series
When you recognize that the terms
1
are
raised to a constant power.
n
Limit test (“Bouncer” test)
If you can easily see how the a n ' s
behave as a sequence.
If lim n→∞ a n ≠ 0 , the series diverges.
If lim n→∞ a n = 0 , the test is inconclusive.
When the series is alternating.
Usually, you’ll see a (−1) n , (−1) n −1 ,
or (−1) n +1 . Sometimes you’ll see a
æ 2n ± 1 ö
sin ç
π ÷ or a cos(nπ )
è 2
ø
Given å (−1) n bn :
IF:
1. bn > 0 for all n AND
2. bn +1 > bn for all n AND
3. lim n→∞ bn = 0
THEN the series converges.
Alternating Series Test
Details
a
ì
converges
to
if r < 1
ï
∞
1
r
−
ï
a⋅rní
å
n =0
ï
diverges if r > 1
ï
î
ìconverges when p > 1
∞
1 ï
å
p í
n =1 n ï
î diverges when p ≤ 1
IF: 3. fails, then the series diverges.
IF: 1. or 2. fail, but 3. Holds, then the test is inconclusive.
Series Tests for Convergence, Page 1/3
Test Name
When to Use
Absolute Convergence
When some terms of the series are
positive, and some are negative, but
the series is not alternating.
IF
åa
Details
n
converges, then
åa
n
converges (and we say that
å a converges absolutely).
IF å a diverges, then the test is inconclusive.
n
n
Ratio Test
When you see factorials
Sometimes, when you see terms to
the nth power.
Do Not Use for rational or algebraic
functions of n (the test will always
come out inconclusive)
Root Test
When the terms are something
raised to the nth power.
Comparison Test
Rational or algebraic functions of n.
Usually compare to a p-series
ì < 1, converges (absolutely)
ï
a ïï
IF lim n→∞ n+1 í
> 1, diverges
an ï
ï
ïî= 1, the test is inconclusive
ì < 1, converges (absolutely)
ï
ïï
IF lim n→∞ n a n í
> 1, diverges
ï
ï
ïî= 1, the test is inconclusive
Can only use when the series
terms!!
Suppose a n ≤ bn for each n.
åa
n
and
å b converges, then so does å a
IF å a diverges, then so does å b
IF
n
n
åb
n
n
.
n
.
have positive
Series Tests for Convergence, Page 2/3
Test Name
Limit Comparison Test
When to Use
Use for comparison when the
comparison test is too vexing.
Details
Can only use when the series
terms!!
åa
n
and
åb
n
have positive
IF
lim n →∞
an
= L AND
bn
AND
L≠0
L≠∞
THEN å a n and
Integral Test
If the integral is easy to evaluate.
åb
n
behave the same way.
IF
∞
∞
n=r
n=r
å a n = å f (n) AND
f ( x) is continuous on [r , ∞) AND
f ( x) > 0 on [r , ∞) AND
f ( x) is decreasing on [r , ∞) ,
THEN
∞
∞
n =r
r
å a n behaves the same way as
ò f ( x)dx .
Series Tests for Convergence, Page 3/3