Download presentation

Presentation is loading. Please wait.

Published byKatelyn Burgess Modified over 3 years ago

1
Solved problems on comparison theorem for series

2
Mika Seppälä: Solved Problems on Comparison Test comparison test Let 0 a k b k for all k.

3
Mika Seppälä: Solved Problems on Comparison Test 1 OVERVIEW OF PROBLEMS Let 0 a k b k for all k. Assume that the series and both converges. Show that the series converges.

4
Mika Seppälä: Solved Problems on Comparison Test 2 OVERVIEW OF PROBLEMS Let a k and b k positive for all k. Assume that the series converges and that Show that the series converges.

5
Mika Seppälä: Solved Problems on Comparison Test 45 67 OVERVIEW OF PROBLEMS Use Comparison Test to determine whether the series converge or diverge. 3

6
Mika Seppälä: Solved Problems on Comparison Test Let 0 a k b k for all k. Assume that the series and both converges. Show that the series converges. Problem 1 COMPARISON TEST

7
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution Since converges,. By definition of limit this means, Assume. Since is positive for all k, we have

8
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) Recall also that by assumptions. Then the Comparison Theorem implies that

9
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) Remark that is suffices to show that

10
Mika Seppälä: Solved Problems on Comparison Test Let a k and b k positive for all k. Assume that the series converges and that Show that the series converges. Problem 2 COMPARISON TEST

11
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution Since, there is a number such that Therefore Since, so does

12
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) This implies by the comparison theorem that

13
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 3 Solution

14
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd)

15
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 4 Solution By rewriting,

16
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) Therefore we can write Hence converges.

17
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 5 Solution

18
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd)

19
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 6 Solution

20
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd)

21
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Problem 7 Solution Since for all n, we obtain

22
Mika Seppälä: Solved Problems on Comparison Test COMPARISON TEST Solution (contd) We know that the geometric series

Similar presentations

Presentation is loading. Please wait....

OK

Convergence or Divergence of Infinite Series

Convergence or Divergence of Infinite Series

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on inhabiting other planets like ours Download free ppt on forest and wildlife resources Ppt on web browser and its features Ppt on 3g mobile technology Ppt on semiconductor diodes solar Ppt online examination project in java Ppt on global warming with sound Ppt on self awareness institute Ppt on digital energy meter Ppt on distributed file system