646 Chapter 9 Infinite Series

SUMMARY OF TESTS FOR SERIES

Test

Term

Geometric Series

Telescoping Series

Series

Alternating Series

Integral( is continuous,positive, anddecreasing)

Root

Ratio

Direct Comparison

Limit Comparisonan, bn > 0

an, bn > 0

f

p-

nth-

Series

n1an

n1an

n1an

n1an

an f n 0

n1an,

n11n1an

n1

1n p

n1bn bn1

n0arn

n1an

Condition(s)of Convergence

and

converges

and converges

and converges

n1bn

limn

anbn

L > 0

n1bn

0 < an bn

limn an1an < 1lim

nnan < 1

1

f x dx

limn

an 00 < an1 an

p > 1

limn

bn L

r < 1

Comment

This test cannot be usedto show convergence.

Sum:

Sum:

Remainder:

Remainder:

Test is inconclusive if

Test is inconclusive if

limn an1an 1.lim

nnan 1.

0 < RN < N

f x dx

RN aN1

S b1 L

S a1 r

Condition(s)of Divergence

diverges

or

or

and diverges

and diverges

n1bn

limn

anbn

L > 0

n1bn

0 < bn an

limn an1an > 1

limn

nan > 1

1

f x dx

0 < p 1

r 1

limn

an 0