115 - the Alternating Series Test. We know that the error...

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MATH 153 Selected Solutions for § 11.5 Exercise 12 : Test the series for convergence or divergence. X n =1 ( - 1) n e 1 /n n Solution : Let b n = e 1 /n n ; then we must show that the b n decrease to zero. First, it is clear that lim n →∞ e 1 /n n = 0 , since e 1 /n 1 as n → ∞ . It remains to show that e 1 / ( n +1) n +1 < e 1 /n n . By rearranging we get e 1 / ( n +1) e 1 /n < n + 1 n , which is clearly true because the left is less than 1 and the right is greater than 1. Thus, by the Alternating Series Test, the series converges. Exercise 18 : Test the series for convergence or divergence. X n =1 ( - 1) n cos ± π n ² Solution : As n → ∞ , we have that lim n →∞ b n = lim n →∞ cos ± π n ² = cos 0 = 1 . Thus, as b n doesn’t go to zero, the series diverges by the Alternating Series Test (or by the Test for Divergence). 1

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2 Exercise 24 : Show that the series is convergent. How many terms of the series do we need to add in order to ﬁnd the sum to the indicated accuracy? X n =1 ( - 1) n n 5 n , | error | < 0 . 0001 Solution : Certainly 1 n 5 n decreases to 0 as n → ∞ , so the series converges by
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Unformatted text preview: the Alternating Series Test. We know that the error of the n th partial sum of a series is bounded by | a n +1 | , so the remaining question boils down to, “for what value of n is 1 n 5 n < . 0001?” 1 n 5 n < . 0001 n 5 n > 10000 n ≥ 5 , since 4 · 5 4 = 2500 and 5 · 5 5 = 15 , 625. Thus, we only need to add the ﬁrst four terms of the series to approximate the sum within the allotted error. Exercise 32 : For what values of p is the following series convergent? ∞ X n =1 (-1) n-1 n p Solution : We apply the Alternating Series Test: for all p > 0, b n = 1 n p → as n → ∞ . Similarly, for all p > 0 we have n p < ( n + 1) p ⇒ 1 ( n + 1) p < 1 n p . Thus, for all p > 0 the series converges by the Alternating Series Test, and for all p ≤ 0 the series diverges by the same....
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This note was uploaded on 07/26/2011 for the course MATH 153 taught by Professor Rempe during the Spring '08 term at Ohio State.

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115 - the Alternating Series Test. We know that the error...

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