Download presentation

Published byJennifer Hale Modified over 4 years ago

1
**Summary of Convergence Tests for Series and Solved Problems**

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

2
**Test Test Quantity Converges if Diverges if Ratio q < 1 q > 1**

Root r < 1 r > 1 Integral Int < ∞ Int = ∞ 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. Mika Seppälä: Series

3
**Comparison Test and the Alternating Series Test**

Assume that 0≤ ak ≤ bk for all k. If the series Alternating Series Test 1 2 Mika Seppälä: Series

4
**Error Estimates Error Estimate by the Integral Test**

Error of the approximation by the Mth partial sum. 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. Mika Seppälä: Series

5
Overview of Problems 1 2 3 Mika Seppälä: Series

6
Overview of Problems 4 5 6 7 8 9 Do the above series 4-5 and 7 – 9 converge or diverge? 10 11 12 Mika Seppälä: Series

7
**Overview of Problems 13 14 15 Do the series in 13 – 16 converge? 16 17**

18 19 20 Do the series in 19 – 20 converge? Mika Seppälä: Series

8
Overview of Problems 21 22 24 25 23 26 27 Do the series given in Problems 23 – 29 converge? 28 29 30 Mika Seppälä: Series

9
Comparison Test 1 Solution Mika Seppälä: Series

10
Comparison Test 2 Solution Mika Seppälä: Series

11
**The Comparison Test 3 Solution The series a) needs not converge.**

Example: Mika Seppälä: Series

12
**The Comparison Test 3 Solution (cont’d) The series b) does converge.**

<1 Mika Seppälä: Series

13
The Integral Test 4 Solution Mika Seppälä: Series

14
**Comparison Test From Applications of Differentiation. 5 Solution**

Mika Seppälä: Series

15
**Partial Fraction Computation**

6 Solution These terms cancel. Mika Seppälä: Series

16
Comparison Test 7 Solution Mika Seppälä: Series

17
**The Integral and the Comparison Tests**

8 Solution Mika Seppälä: Series

18
**The Integral Test 9 Solution**

Hence the series diverges by the Integral Test. Mika Seppälä: Series

19
**The Integral Test 10 Solution**

Computing 1000th partial sum by Maple we get the approximation The precise value of the above infinite sum is π2/6≈ Mika Seppälä: Series

20
**The Comparison Test 11 Solution**

You can show this also directly by the Integral Test without referring to the Harmonic Series. Mika Seppälä: Series

21
The Comparison Test 12 Solution Mika Seppälä: Series

22
**The Alternating Series Test**

13 Solution Mika Seppälä: Series

23
**The Alternating Series Test**

14 Solution Mika Seppälä: Series

24
**The Alternating Series Test**

15 Solution Mika Seppälä: Series

25
**The Alternating Series Test**

16 Solution This follows from the fact that the sine function is increasing for 0≤x≤π/2. Mika Seppälä: Series

26
**The Alternating Series Test**

17 Solution Mika Seppälä: Series

27
**The Alternating Series Test**

18 Solution Mika Seppälä: Series

28
**The Alternating Series Test**

19 19 Solution Use l’Hospital’s Rule. Mika Seppälä: Series

29
**The Alternating Series Test**

20 Solution Use l’Hospital’s Rule. Mika Seppälä: Series

30
The Integral Test 21 Solution Mika Seppälä: Series

31
**The Integral Test 22 Solution**

This requires that p≠1. If p=1, the corresponding improper integral diverges. Mika Seppälä: Series

32
The Root Test 23 Solution Use the Root Test. Mika Seppälä: Series

33
The Ratio Test 24 Solution Use the Ratio Test. Mika Seppälä: Series

34
**The Comparison Test 25 Solution Use the Comparison Test.**

According to Problem 21. Conclude that the series converges. Mika Seppälä: Series

35
The Ratio Test 26 Solution Use the Ratio Test. Mika Seppälä: Series

36
The Ratio Test 27 Solution Use the Ratio Test. Mika Seppälä: Series

37
**The Ratio Test 28 Solution**

Conclude that the series converges by the Ratio Test. Mika Seppälä: Series

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. Mika Seppälä: Series

39
The Ratio Test 30 Solution Use the Ratio Test. Mika Seppälä: Series

Similar presentations

OK

DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

DIVIDING INTEGERS 1. IF THE SIGNS ARE THE SAME THE ANSWER IS POSITIVE 2. IF THE SIGNS ARE DIFFERENT THE ANSWER IS NEGATIVE.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on object-oriented programming vs procedural programming Ppt on water transport in india Eat before dentist appt on your birthday Ppt on hvdc system in india Ppt on data backup and recovery Best ppt on save water Ppt on phonetic transcription Ppt on 9/11 terror attack impact on world economy Ppt on work and energy class 9 Ppt on electricity for class 10th board