homework5solnmat25

homework5solnmat25 - Homework 5 Solutions 1(a Use induction...

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Homework 5 Solutions 1. (a) Use induction to prove the following formula for all n 1: 1 3 + 1 15 + · · · + 1 4 n 2 - 1 = n 2 n + 1 Solution: When n = 1, the equality 1 3 = 1 3 holds, so the base case is proved. Now, suppose 1 3 + 1 15 + · · · + 1 4 k 2 - 1 = k 2 k +1 . Then 1 3 + 1 15 + · · · + 1 4( k + 1) 2 - 1 = k 2 k + 1 + 1 4( k + 1) 2 - 1 = k 2 k + 1 + 1 (2 k + 1)(2 k + 3) = (2 k + 3) k + 1 (2 k + 1)(2 k + 3) = k + 1 2( k + 1) + 1 (b) Find the sum n =1 1 4 n 2 - 1 . Solution: X n =1 1 4 n 2 - 1 = lim n 2 n + 1 = 1 2 2. Determine which of the following series converge and which diverge. Justify your answers. (a) cos( nπ/ 2). Solution: The series diverges because cos( nπ/ 2) does not converge to 0. (b) 1 n ( n - 1) . Solution: The series converges. It can be rewritten as a telescoping series: 1 2 + 1 3 · 2 + · · · + 1 n ( n - 1) = 1 - 1 2 + 1 2 - 1 3 + · · · + 1 n - 1 - 1 n = 1 - 1 n 1 Note: there are plenty of other ways to prove it converges. (c) 3 n e - n . Solution: The series diverges becausue 3 e > 1 so ( 3 e ) n diverges to . (d) 1 ln n . Solution: The series diverges by the comparison test because 1 ln n > 1 n , and we know the harmonic series 1 n diverges.
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