Summary of Convergence Tests for Series and Solved Problems

1
Summary of Convergence Tests for Series and
Solved Problems

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

2
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.
3
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
4
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.
5
Overview of Problems
1
2
3
6
Overview of Problems
6
4
5
7
8
9
Do the above series 4-5 and 7 9 converge or
diverge
10
11
12
7
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
8
Overview of Problems
21
22
25
24
23
27
26
Do the series given in Problems 23 29 converge
28
29
30
9
Comparison Test
1
Solution
10
Comparison Test
2
Solution
11
The Comparison Test
3
The series a) needs not converge.
Solution
Example
12
The Comparison Test
3
The series b) does converge.
Solution (contd)
13
The Integral Test
4
Solution
14
Comparison Test
5
Solution
15
Partial Fraction Computation
6
Solution
16
Comparison Test
7
Solution
17
The Integral and the Comparison Tests
8
Solution
18
The Integral Test
9
Solution
Hence the series diverges by the Integral Test.
19
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.
20
The Comparison Test
11
Solution
21
The Comparison Test
12
Solution
22
The Alternating Series Test
13
Solution
23
The Alternating Series Test
14
Solution
24
The Alternating Series Test
15
Solution
25
The Alternating Series Test
16
Solution
26
The Alternating Series Test
17
Solution
27
The Alternating Series Test
18
Solution
28
The Alternating Series Test
19
19
Solution
29
The Alternating Series Test
20
Solution
30
The Integral Test
21
Solution
31
The Integral Test
22
Solution
This requires that p1. If p1 the corresponding
improper integral diverges.
32
The Root Test
23
Solution
Use the Root Test.
33
The Ratio Test
24
Solution
Use the Ratio Test.
34
The Comparison Test
25
Solution
Use the Comparison Test.
According to Problem 21.
Conclude that the series converges.
35
The Ratio Test
26
Solution
Use the Ratio Test.
36
The Ratio Test
27
Solution
Use the Ratio Test.
37
The Ratio Test
28
Solution
Conclude that the series converges by the Ratio
Test.
38
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.
39
The Ratio Test
30
Solution
Use the Ratio Test.