Unformatted text preview: of the problem. 2. If the partial sums s n of ∑ ∞ n =1 a n are bounded, show that the series ∑ ∞ n =1 a n /n converges to ∑ ∞ n =1 s n /n ( n + 1). Here we apply Abel’s Lemma. We then see that m X n =1 a n n = 1 m s m + m1 X n =1 ² 1 n1 n + 1 ³ s n = s m m + m1 X n =1 1 n ( n + 1) s n . As m → ∞ , we have, by the Pinching principle, s m m → 0 since s m is bounded and 1 /m → 0. What remains after taking this limit is the desired equality. 1...
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 Spring '10
 MetCalfe
 Calculus, Geometric Series, Mathematical Series, Negative and nonnegative numbers, partial sums sn

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