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HW5 - doesn’t use Theorem 2.4.6 Recall that we already...

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Hints for Exercise 2.7.1, Part (c) 1. First show that the partial sums s n are ordered as follows (see the picture on page 35): s 2 < s 2 < · · · < s 2 n < · · · < s 2 n +1 < · · · < s 3 < s 1 . In other words the even-numbered partial sums form an increasing sequence, the odd-numbered partial sums form a decreasing sequence, and the even-numbered ones are less than the odd-numbered ones. 2. Show that ( s 2 n ) and ( s 2 n +1 ) are both convergent sequences. 3. Show that lim s 2 n = lim s 2 n +1 . (Hint: what can you say about the difference of these two limits?) 4. Show that ( s n ) converges, and explain why this proves Theorem 2.7.7. Problem A In this problem you are going to prove that the p -series n =1 1 n p converges if and only if p > 1. (This is stated as Corollary 2.4.7 in the book, intending the proof to be based on Theorem 2.4.6 which we are not covering. This problem leads you through a different proof which
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Unformatted text preview: doesn’t use Theorem 2.4.6.) Recall that we already know from our discussion of Section 2.4 that the series is divergent for p = 1 and convergent for p = 2. (a) Suppose p < p . Use the Comparison Test to show that if ∑ 1 /n p is convergent then ∑ 1 /n p is also convergent. (b) Show that the p-series diverges for any p ≤ 1. (c) Consider p > 1. (1) Define the sequence ( b n ) by b 1 = 1, and b n = Z n n-1 1 x p dx for n ≥ 1 . Explain why 1 n p ≤ b n . (You are free to use properties of integrals that you know from calculus.) (2) The partial sums for ∑ ∞ n =1 b n are s n = b 1 + ··· + b n = 1 + Z n 1 1 x p dx. Show that s n ≤ 1 + 1 p-1 . (Just calculate the integral!) (3) Explain why ( s n ) and ∑ ∞ n =1 b n both converge. (4) Conclude that ∑ ∞ n =1 1 /n p converges....
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