Math 140A Test 2 100 points November 23, 2009 Professor Evans Directions: Show all work. In your proofs, state where you use the hypotheses. Notation: Throughout, z and a n ( n = 0 , 1 , 2 , 3 ,... ) are complex. (If you can only handle the real case, you still get considerable partial credit.) Points: Each problem is worth 20 points. (1) Given that a n → 0, show that ( a 1 + ··· + a n ) /n → 0, as n → ∞ . Solution: Let ± > 0. Let s n = a 1 + ··· + a n . There exists an integer N such that | a n | < ± for all n > N . There exists a positive constant M (independent of ± ) for which | a n | < M for all n ≤ N . Thus by the triangle inequality, | s n | ≤ NM + ( n-N ) ± . Thus | s n | /n < NM/n + ± , so | s n | /n < 2 ± for all n > NM/± . This shows the desired result that s n /n → 0. (2) Consider the power series ∑ ∞ n =0 z n / ( n + 1). Determine the values of z for which the series (A) diverges (B) converges. Justify brieﬂy. Solution: It’s easy to see using the ratio test or the root test that the series diverges for
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