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Quiz6 Solutions - MAC 2312 Quiz 6 SOLUTIONS x 1 Determine...

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MAC 2312 Quiz 6 February 28, 2012 SOLUTIONS 1. Determine the nonzero values of x for which the series n =1 x 5 x 2 n - 1 converges. Solution: First, observe that the numerator of the summand is independent of n , so summationdisplay n =1 x 5 x 2 n 1 = x 5 summationdisplay n =1 1 x 2 n 1 . Therefore, we only need to find the values of x for which the sum on the right-hand side converges. To achieve this, consider two cases separately. Case 1: | x | < 1 In this case, lim n →∞ x 2 n = 0, so the limit of the summand is lim n →∞ 1 x 2 n 1 = 1 negationslash = 0 . Hence, by the test for divergence of series, n =1 1 x 2 n - 1 is divergent. Case 2: | x | > 1 In this case, lim n →∞ x 2 n = + , so lim n →∞ 1 x 2 n 1 x 2 n - 1 = lim n →∞ x 2 n 1 x 2 n = 1 . Therefore, by the limit comparison test, n =1 1 x 2 n - 1 converges if and only if n =1 1 x 2 n converges. But since | x | > 1, 1 x 2 < 1. Therefore in view of the geometric series, summationdisplay n =1 1 x 2 n = summationdisplay n =1 parenleftbigg 1 x 2 parenrightbigg n < , so n - 1 1 x 2 n - 1 converges for | x | > 1. 2. Determine the positive values of k for which the series n =2 n n - n k converges. Solution: Here is the intuition behind finding the solution (this is not a solution; it is just a way to come up with a candidate for a solution). If k > 1, then as n → ∞ , n k dominates n , so the summand is approximately n n n k n n k = 1 n k - 1 / 2 . Therefore, by the p -series test, we suspect that the given sum will converge if and only if k > 3 / 2 (because this makes k 1 2 > 1).
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With a candidate for a solution in hand (i.e. k > 3 2 ), we are ready to check whether our proposed solution is an actual solution. This amounts to performing two com- putations. First, we must show that the given series converges if k > 3 2 . Second, we
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