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

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Mika Seppälä: Series Comparison Test and the Alternating Series Test Comparison Test Assume that 0 a k b k for all k. If the series Alternating Series Test 1 2

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Mika Seppälä: Series Error Estimates Error Estimate by the Integral Test Error Estimates by the Alternating Series Test Error of the approximation by the M th partial sum. 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.

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Mika Seppälä: Series Overview of Problems 2 1 3

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

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Mika Seppälä: Series Overview of Problems Do the series in 13 – 16 converge? Do the series in 19 – 20 converge?

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Mika Seppälä: Series Overview of Problems Do the series given in Problems 23 – 29 converge?

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Mika Seppälä: Series Comparison Test 1 Solution

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Mika Seppälä: Series Comparison Test 2 Solution

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Mika Seppälä: Series The Comparison Test 3 Solution The series a) needs not converge. Example:

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Mika Seppälä: Series The Comparison Test 3 Solution (contd) The series b) does converge. <1

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Mika Seppälä: Series The Integral Test 4 Solution

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Mika Seppälä: Series Comparison Test 5 Solution From Applications of Differentiation.

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Mika Seppälä: Series Partial Fraction Computation 6 Solution These terms cancel.

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Mika Seppälä: Series Comparison Test 7 Solution

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Mika Seppälä: Series The Integral and the Comparison Tests 8 Solution

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Mika Seppälä: Series The Integral Test 9 Solution Hence the series diverges by the Integral Test.

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Mika Seppälä: Series The Integral Test 10 Solution Computing 1000 th partial sum by Maple we get the approximation The precise value of the above infinite sum is π 2 /

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Mika Seppälä: Series The Comparison Test Solution You can show this also directly by the Integral Test without referring to the Harmonic Series. 11

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Mika Seppälä: Series The Comparison Test 12 Solution

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Mika Seppälä: Series The Alternating Series Test 13 Solution

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Mika Seppälä: Series The Alternating Series Test 14 Solution

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Mika Seppälä: Series The Alternating Series Test 15 Solution

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Mika Seppälä: Series The Alternating Series Test 16 Solution This follows from the fact that the sine function is increasing for 0xπ/2.

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Mika Seppälä: Series The Alternating Series Test 17 Solution

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Mika Seppälä: Series The Alternating Series Test 18 Solution

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Mika Seppälä: Series The Alternating Series Test 19 Solution Use lHospitals Rule.lHospitals Rule 19

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Mika Seppälä: Series The Alternating Series Test 20 Solution Use lHospitals Rule.lHospitals Rule

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Mika Seppälä: Series The Integral Test 21 Solution

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Mika Seppälä: Series The Integral Test 22 Solution This requires that p1. If p=1, the corresponding improper integral diverges.

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

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

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Mika Seppälä: Series The Comparison Test 25 Solution Use the Comparison Test. According to Problem 21. Conclude that the series converges.

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

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

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Mika Seppälä: Series The Ratio Test 28 Solution Conclude that the series converges by the Ratio Test.

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Mika Seppälä: Series The Ratio Test 29 Solution The series converges by the Ratio Test. Observe that for all positive integers n, sin(n) + cos(n) 0. Hence, for every n, a n 0, and the above ratio is defined for all n.

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

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