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The Comparison Test Let 0 a k b k for all k.

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Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges, then also the series converges, and

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Mika Seppälä The Comparison Test Claim converges. Observe that the partial sums S m = a 1 + a 2 + … + a m form an increasing sequence since a k 0 for all k. Proof

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Mika Seppälä The Comparison TEST Claim converges. The assumptions imply Observe that the sum is finite since this series converges. Proof (contd)

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Mika Seppälä The Comparison test Claim converges. Proof (contd) The partial sums form a bounded increasing sequence. Hence the limit exists and is finite.

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Mika Seppälä The Comparison test Comparison Theorem B Assume that 0 a k b k for all k. If the series diverges, then also the series diverges.

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Mika Seppälä The Comparison Test Claim diverges. Since the series diverges, the partial sums S m form an unbounded set. The assumptions imply Proof

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Mika Seppälä Claim diverges. Hence Proof (contd) Since, also THE COMPARISON TEST

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Mika Seppälä Example Show that the series converges. For all integer values of k, 1 < 2 + sin k < 3. Solution Hence for all integer values of k. THE COMPARISON TEST

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Mika Seppälä Example Show that the series converges. The series is a convergent geometric series Solution (contd) Hence converges by the Comparison Theorem A. THE COMPARISON TEST

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Mika Seppälä Example Show that the series diverges. Hence for all positive integer values of k. For all positive integers k, Solution THE COMPARISON TEST

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Mika Seppälä Example Show that the series diverges. Since the Harmonic Series diverges, also Solution (contd) diverges by the Comparison Theorem B. THE COMPARISON TEST

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The Comparison Test Let 0 a k b k for all k.

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