1

Convergence or Divergence of Infinite Series

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Let be an infinite series of positive terms.

The series converges if and only if the sequence of partial sums,

, converges. This means:

Definition of Convergence for an infinite series:

1nna

aaaaSn 321

1

limn

nnn

aS

If , the series diverges.

Divergence Test:0lim

nna

1nna

Example: The series

12 1n n

nis divergent since 1

11

1lim

1lim

22

nn

nnn

However, 0lim n

na does not imply convergence!

3

Special Series

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Geometric Series:

The Geometric Series: converges for

If the series converges, the sum of the series is:

Example: The series with and converges .

The sum of the series is 35.

 

12 nararara 11 r

r

a

1

n

n

1 8

75

8

351 aa

8

7r

5

p-Series:

The Series: (called a p-series) converges for

and diverges for

Example: The series is convergent. The series is divergent.

 

1p

1

1

npn1p

1001.1

1

n n

1

1

n n

6

Convergence Tests

11.3 The Integral Test

11.4 The Comparison Tests

11.5 Alternating Series

11.6 Ratio and Root Tests

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Integral test:

If f is a continuous, positive, decreasing function on with

then the series converges if and only if the improper integral converges.

Example: Try the series: Note: in general for a series of the form:

13

1

n n

),1[ nanf )(

1nna

1

)( dxxf

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Comparison test:

If the series and are two series with positive terms, then:

(a)If is convergent and for all n, then converges.

(b)If is divergent and for all n, then diverges.

• (smaller than convergent is convergent)• (bigger than divergent is divergent)

Examples: which is a divergent harmonic series. Since the original series is larger by comparison, it is divergent.

which is a convergent p-series. Since the

original series is smaller by comparison, it is

convergent.

1nna

1nnb

1nnb nn ba

1nna

1nnb nn ba

1nna

22

22

2

13

3

2

3

nnn nn

n

n

n

1

21

31

23

1

2

5

2

5

12

5

nnn nn

n

nn

n

10

Limit Comparison test:

If the the series and are two series with positive terms, and if

where then either both series converge or both series diverge.

Useful trick: To obtain a series for comparison, omit lower order terms in the numerator and the denominator and then simplify.

Examples: For the series compare to which is a convergent p-series.

For the series compare to which is a divergent geometric series.

1nna

1nnb c

b

a

n

n

n

lim

c0

12 3n nn

n

1 2

31

2

1

nn nn

n

123n

n

n

n

n

11 33 n

n

nn

n

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Alternating Series test:

If the alternating series satisfies:

and then the series converges.

Definition: Absolute convergence means that the series converges without alternating (all signs and terms are positive).

Example: The series is convergent but not absolutely convergent.

Alternating p-series converges for p > 0.  

Example: The series and the Alternating Harmonic series are convergent.

1

65432111

nn

n bbbbbbb

1 nn bb 0lim n

nb

0 1

1

n

n

n

1

)1(

np

n

n

1

)1(

n

n

n

1

)1(

n

n

n

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Ratio test:

(a)If then the series converges;

(b)If the series diverges.

(c)Otherwise, you must use a different test for convergence.

If this limit is 1, the test is inconclusive and a different test is required.

Specifically, the Ratio Test does not work for p-series.

Example:

1lim 1

n

n

n a

a

1nna

1lim 1

n

n

n a

a

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11.7 Strategy for Testing Series

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Summary:Apply the following steps when testing for convergence:

1.Does the nth term approach zero as n approaches infinity? If not, the Divergence Test implies the series diverges.

2.Is the series one of the special types - geometric, telescoping, p-series, alternating series?

3.Can the integral test, ratio test, or root test be applied?

4.Can the series be compared in a useful way to one of the special types?

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Summaryof all tests:

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Example

Determine whether the series converges or diverges using the Integral Test.

Solution: Integral Test:

Since this improper integral is divergent, the series

(ln n)/n is also divergent by the Integral Test.

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More examples

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Example 1

Since as an ≠ as n , we should use the Test for Divergence.

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Example 2: Using the limit comparison test

Since an is an algebraic function of n, we compare the given series with a p-series.

The comparison series for the Limit Comparison Test is

where

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Example 3

Since the series is alternating, we use the Alternating Series Test.

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Example 4

Since the series involves k!, we use the Ratio Test.

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Example 5

Since the series is closely related to the geometric series

, we use the Comparison Test.