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Summary of Convergence Tests for Series and

Solved Problems

- Integral Test
- Ratio Test
- Root Test
- Comparison Theorem for Series
- Alternating Series

The above test quantities can be used to study

the convergence of the series S.

In the Integral Test we assume that there is a

decreasing non-negative function f such that

ak f(k) for all k. The Test Quantity of the

Integral Test is the improper integral of this

function.

Comparison Test and the Alternating Series Test

Comparison Test

Assume that 0 ak bk for all k. If the

series

Alternating Series Test

1

2

Error Estimates

Error Estimate by the Integral Test

Error Estimates by the Alternating Series Test

This means that the error when estimating the sum

of a converging alternating series is at most the

absolute value of the first term left out.

Overview of Problems

1

2

3

Overview of Problems

6

4

5

7

8

9

Do the above series 4-5 and 7 9 converge or

diverge?

10

11

12

Overview of Problems

15

13

14

Do the series in 13 16 converge?

16

17

18

20

19

Do the series in 19 20 converge?

Overview of Problems

21

22

25

24

23

27

26

Do the series given in Problems 23 29 converge?

28

29

30

Comparison Test

1

Solution

Comparison Test

2

Solution

The Comparison Test

3

The series a) needs not converge.

Solution

Example

The Comparison Test

3

The series b) does converge.

Solution (contd)

The Integral Test

4

Solution

Comparison Test

5

Solution

Partial Fraction Computation

6

Solution

Comparison Test

7

Solution

The Integral and the Comparison Tests

8

Solution

The Integral Test

9

Solution

Hence the series diverges by the Integral Test.

The Integral Test

10

Solution

Computing 1000th partial sum by Maple we get the

approximation 1.6439. The precise value of the

above infinite sum is p2/61.6449.

The Comparison Test

11

Solution

The Comparison Test

12

Solution

The Alternating Series Test

13

Solution

The Alternating Series Test

14

Solution

The Alternating Series Test

15

Solution

The Alternating Series Test

16

Solution

The Alternating Series Test

17

Solution

The Alternating Series Test

18

Solution

The Alternating Series Test

19

19

Solution

The Alternating Series Test

20

Solution

The Integral Test

21

Solution

The Integral Test

22

Solution

This requires that p?1. If p1, the corresponding

improper integral diverges.

The Root Test

23

Solution

Use the Root Test.

The Ratio Test

24

Solution

Use the Ratio Test.

The Comparison Test

25

Solution

Use the Comparison Test.

According to Problem 21.

Conclude that the series converges.

The Ratio Test

26

Solution

Use the Ratio Test.

The Ratio Test

27

Solution

Use the Ratio Test.

The Ratio Test

28

Solution

Conclude that the series converges by the Ratio

Test.

The Ratio Test

29

Solution

Observe that for all positive integers n, sin(n)

cos(n) ?0. Hence, for every n, an? 0, and the

above ratio is defined for all n.

The series converges by the Ratio Test.

The Ratio Test

30

Solution

Use the Ratio Test.

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