Calculus 2 Cheat Sheet Page 14
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16B Veitch
Calculus 2 Study Guide
12. Summary of Convergence Tests
Test
Theorem
Comments
Divergence
Given the series
a
, if lim
a
= 0, the series
Can only be used to test for divergence. It
n
n
n
Test
diverges.
can never be used to test for convergence.
1
P-Test
The series
converges if p > 1. If p
1,
You’re normally using this series as a com-
p
n
n=1
parison for Direct or Limit Comparison
1
then
diverges.
p
Tests
n
n=1
Integral
Given the series
a
, if you can write a
= f (n)
The function f should be easily integrated
n
n
Test
using one of our integration techniques
and
f (n) dn converges, then so does
a
.
n
1
from previous chapters. This test should
If
f (n) dn diverges, so does
a
n
not be used when a
has factorials.
1
n
Direct Com-
Let
a
and
b
be series such that 0
a
You need to be able to anticipate the con-
n
n
n
parison Test
b
. If
b
converges, so does
a
. If
a
vergence. This allows you to choose an ap-
n
n
n
n
diverges, so does
b
.
propriate comparison. You should know
n
the convergence of one of the series.
Limit Com-
Let
a
and
b
be series where a
0 and
Can be easier than Direct Comparison
n
n
n
a
n
parison Test
b
0. If lim
= L, where L is a positive
Test since getting the required inequali-
n
b
n
n
finite number, then either both series converge or
ties to work can be time consuming.
they both diverge.
n
Alternating
Given the series
( 1)
b
, if b
> 0, decreasing,
You have to check for absolute or
n
n
Series Test
and b
0, then the series converges.
conditional
convergence
by
testing
n
n
( 1)
b
=
b
.
n
n
a
n+1
Ratio Test
Let
a
be a series. Let L = lim
.
Use this test when you have exponential
n
a
n
n
If L < 1, the series converges absolutely.
functions or factorials.
If L > 1, the series diverges.
If L = 1, the test is inconclusive.
n
Root Test
Let
a
be a series. Let L = lim
a
.
Use this test when a
= (b
)
.
n
n
n
n
n
If L < 1, the series converges absolutely.
If L > 1, the series diverges.
If L = 1, the test is inconclusive.
Absolute
If the series
a
converges absolutely, then it
If
a
converges, when
a
is abso-
n
n
n
Conver-
converges.
lutely convergent.
gence
Conditional
If the series
a
converges but
a
diverges,
You’re usually checking if
a
con-
n
n
n
Conver-
then the series
a
converges conditionally.
verges because
a
converged by AST
n
n
gence
14
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