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 pseries diverges for any p ≤ 1. (c) Consider p > 1. (1) Deﬁne the sequence ( b n ) by b 1 = 1, and b n = Z n n1 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 p1 . (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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 '08
 KBHANNSGEN
 Calculus, Limit of a sequence, partial sums

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