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Lecture 5

This

**preview**shows half of the first page. to view the full**1 pages of the document.**By Saeid Samiezadeh

This document is intended to summarize various tests used to determine convergence/divergence of series

in Calculus II (MTH 240). It CANNOT be used in anyway as an aid in closed-book tests/quizzes.

Series or Test Form of Series Condition for

Convergence Condition for

Divergence Comments

Geometric Series 0

n

n

ar

0a 1r

1r If convergent,

01

n

n

a

ar r

Useful for comparison tests

Divergence Test 1n

n

a

Does not apply lim 0

n

na

Cannot be used to prove

convergence

Integral Test

1n

n

a

where

()

n

afnand f is

continuous, positive,

and decreasing

1()

f

xdx

converges 1()

f

xdx

diverges

The value of the integral is not

the value of the series.

P-series 1

1

p

nn

1p 1p

Useful for comparison tests

Telescopic Series

1nn

n

ab

whose

partial sums ( n

s)

only have a fixed

number of terms

after cancellation

lim n

ns

lim n

ns

The sum of the series can be

found by lim n

ns

.

Direct

Comparison Test 1n

n

a

where 0

n

a

and 0

n

b

nn

ab

and

1n

n

b

converges

nn

ba

and

1n

n

b

diverges 1n

n

a

is given; you supply

1n

n

b

The test determines absolute

convergence.

Limit

Comparison Test 1n

n

a

where 0

n

a

and 0

n

b

0lim

n

nn

a

b

and

1n

n

b

converges

0lim

n

nn

a

b

and

1n

n

b

diverges

1n

n

a

is given; you supply

1n

n

b

The test determines absolute

convergence.

Alternating

Series Test 1

(1)

nn

n

b

n

bis non-

increasing and

lim 0

n

nb

Does not

apply

If lim 0

n

nb

, divergence test

shall be used.

Ratio Test 1n

n

a

1

lim 1

n

nn

a

a

1

lim 1

n

nn

a

a

Inconclusive if 1

lim 1

n

nn

a

a

Root Test 1n

n

a

lim 1

nn

na

lim 1

nn

na

Inconclusive if

lim 1

nn

na

Absolute

Convergence 1n

n

a

1n

n

a

converges

Does not

apply Applies to arbitrary series

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