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Published byJennifer Hale Modified over 4 years ago

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**Summary of Convergence Tests for Series and Solved Problems**

Integral Test Ratio Test Root Test Comparison Theorem for Series Alternating Series

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**Test Test Quantity Converges if Diverges if Ratio q < 1 q > 1**

Root r < 1 r > 1 Integral Int < ∞ Int = ∞ The above test quantities can be used to study the convergence of the series S. In the Integral Test we assume that there is a decreasing non-negative function f such that ak = f(k) for all k. The Test Quantity of the Integral Test is the improper integral of this function. Mika Seppälä: Series

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**Comparison Test and the Alternating Series Test**

Assume that 0≤ ak ≤ bk for all k. If the series Alternating Series Test 1 2 Mika Seppälä: Series

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**Error Estimates Error Estimate by the Integral Test**

Error of the approximation by the Mth partial sum. Error Estimates by the Alternating Series Test This means that the error when estimating the sum of a converging alternating series is at most the absolute value of the first term left out. Mika Seppälä: Series

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Overview of Problems 1 2 3 Mika Seppälä: Series

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Overview of Problems 4 5 6 7 8 9 Do the above series 4-5 and 7 – 9 converge or diverge? 10 11 12 Mika Seppälä: Series

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**Overview of Problems 13 14 15 Do the series in 13 – 16 converge? 16 17**

18 19 20 Do the series in 19 – 20 converge? Mika Seppälä: Series

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Overview of Problems 21 22 24 25 23 26 27 Do the series given in Problems 23 – 29 converge? 28 29 30 Mika Seppälä: Series

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Comparison Test 1 Solution Mika Seppälä: Series

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Comparison Test 2 Solution Mika Seppälä: Series

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**The Comparison Test 3 Solution The series a) needs not converge.**

Example: Mika Seppälä: Series

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**The Comparison Test 3 Solution (cont’d) The series b) does converge.**

<1 Mika Seppälä: Series

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The Integral Test 4 Solution Mika Seppälä: Series

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**Comparison Test From Applications of Differentiation. 5 Solution**

Mika Seppälä: Series

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**Partial Fraction Computation**

6 Solution These terms cancel. Mika Seppälä: Series

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Comparison Test 7 Solution Mika Seppälä: Series

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**The Integral and the Comparison Tests**

8 Solution Mika Seppälä: Series

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**The Integral Test 9 Solution**

Hence the series diverges by the Integral Test. Mika Seppälä: Series

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**The Integral Test 10 Solution**

Computing 1000th partial sum by Maple we get the approximation The precise value of the above infinite sum is π2/6≈ Mika Seppälä: Series

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**The Comparison Test 11 Solution**

You can show this also directly by the Integral Test without referring to the Harmonic Series. Mika Seppälä: Series

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The Comparison Test 12 Solution Mika Seppälä: Series

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**The Alternating Series Test**

13 Solution Mika Seppälä: Series

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**The Alternating Series Test**

14 Solution Mika Seppälä: Series

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**The Alternating Series Test**

15 Solution Mika Seppälä: Series

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**The Alternating Series Test**

16 Solution This follows from the fact that the sine function is increasing for 0≤x≤π/2. Mika Seppälä: Series

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**The Alternating Series Test**

17 Solution Mika Seppälä: Series

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**The Alternating Series Test**

18 Solution Mika Seppälä: Series

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**The Alternating Series Test**

19 19 Solution Use l’Hospital’s Rule. Mika Seppälä: Series

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**The Alternating Series Test**

20 Solution Use l’Hospital’s Rule. Mika Seppälä: Series

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The Integral Test 21 Solution Mika Seppälä: Series

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**The Integral Test 22 Solution**

This requires that p≠1. If p=1, the corresponding improper integral diverges. Mika Seppälä: Series

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The Root Test 23 Solution Use the Root Test. Mika Seppälä: Series

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The Ratio Test 24 Solution Use the Ratio Test. Mika Seppälä: Series

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**The Comparison Test 25 Solution Use the Comparison Test.**

According to Problem 21. Conclude that the series converges. Mika Seppälä: Series

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The Ratio Test 26 Solution Use the Ratio Test. Mika Seppälä: Series

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The Ratio Test 27 Solution Use the Ratio Test. Mika Seppälä: Series

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**The Ratio Test 28 Solution**

Conclude that the series converges by the Ratio Test. Mika Seppälä: Series

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**The Ratio Test 29 Solution**

Observe that for all positive integers n, sin(n) + cos(n) ≠0. Hence, for every n, an≠ 0, and the above ratio is defined for all n. The series converges by the Ratio Test. Mika Seppälä: Series

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The Ratio Test 30 Solution Use the Ratio Test. Mika Seppälä: Series

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