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Solved problems on comparison theorem for series

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

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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.

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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.

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

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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

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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

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

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

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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

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

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

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

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

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

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

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

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

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

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

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

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

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