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Solved problems on comparison theorem for series.

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Presentation on theme: "Solved problems on comparison theorem for series."— Presentation transcript:

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 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


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