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Goal: Does a series converge or diverge? Lecture 24 – Divergence Test 1 Divergence Test (If a series converges, then sequence converges to 0.)
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Example 1 – Converge/Diverge? 2 Example 2 – Converge/Diverge?
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Example 3 – Converge/Diverge? 3 However,
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Example 4 – Converge/Diverge? 4 However,
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Goal: Does a series converge or diverge? Lecture 25 – Integral Test 5 Integral Test (The area under a function and infinite sum of the terms in a sequence defined by that function are related.)
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6 If area under curve is bounded, then so is But then is a bounded, monotonic sequence. So it converges and thus converges. If area under curve is unbounded, then is also unbounded. And thus, diverges.
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Example 1 – Converge/Diverge? 7 sequence converges to zero. No info from Divergence Test.
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Example 2 – Converge/Diverge? 8 sequence converges to zero. No info from Divergence Test.
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Example 3 – Converge/Diverge? 9 sequence converges to zero. No info from Divergence Test.
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Example 4 – Converge/Diverge? 10 sequence converges to zero. No info from Divergence Test.
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11 Lecture 26 – Ratio and Root Tests Goal: Does a series of positive terms converge or diverge? Ratio Test (Does ratio of successive terms approach some limit L? Then series is close to being geometric.)
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12 Root Test (Does n th root of terms approach some limit L? Then series is close to being geometric.)
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Example 1 – Converge/Diverge? 13 sequence converges to zero. No info from Divergence Test.
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Example 2 14 For what values k does the series converge?
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Example 3 – Converge/Diverge? 15 sequence converges to zero. No info from Divergence Test.
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Example 4 – Converge/Diverge? 16
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1. 2. Direct Comparison: 17 Lecture 27 – Comparison Tests
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Limit Comparison: 18
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Example 1 – Converge/Diverge? 19 sequence converges to zero. No info from Divergence Test.
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Example 2 – Converge/Diverge? 20 sequence converges to zero. No info from Divergence Test.
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Example 3 – Converge/Diverge? 21 sequence converges to zero. No info from Divergence Test.
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Example 4 – Converge/Diverge? 22 sequence converges to zero. No info from Divergence Test.
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