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Calculus and Beyond Homework Help
Value of an infinite sum
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[QUOTE="Ray Vickson, post: 5945314, member: 330118"] Let ##H(N) = \sum_{n=1}^N 1/n## be the Harmonic function; see, eg., [URL]https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)[/URL] Let $$S(N) = \sum_{k=1}^N 1/(2k^2-k).$$ Expand ##1/(2k^2-k)## into partial fractions and note that one of the parts of ##S(N)## involves the sum of the odd reciprocals from ##1## to ##2N-1##, that is ##S_{\text{odd}}##: $$S_{\text{odd}} =\sum_{k=1}^N 1/(2k-1).$$ This is the sum of all reciprocals from 1 to ##2N##, less the sum of the even reciprocals from 2 to ##2N##; the latter can be expressed in terms of ##H(N)##. Thus, we can express ##S(N)## exactly in terms of ##H(N)## and ##H(2N)##. The limit as ##N \to \infty## is then do-able. [/QUOTE]
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Value of an infinite sum
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