everyone! I just would like to know how you can calculate $$\sum_{n=1}^\infty \frac{1}{n\times2^n}$$ I know this does converge, but I don't know how to calculate it numerically.
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Welcome to Mathematics Stack Exchange. You must mean $\sum\limits_{\color{red}n=1}^\infty$, and it's $\ln 2$ – J. W. Tanner Feb 23 '20 at 04:02
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1Cf. this question – J. W. Tanner Feb 23 '20 at 04:22
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Yes! Thank you @J.W.Tanner. I've edited my question. – kim Feb 23 '20 at 11:38
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Does this answer your question? Compute the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$ – rtybase Feb 23 '20 at 11:41
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1Yes! Thanks! @rtybase – kim Feb 23 '20 at 11:43
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Hint:
$\ln(1+x)=x-\dfrac{x^2}2+\dfrac{x^3}3-\dfrac{x^4}4+...$
Now take $x=-\frac{1}2$ and note that $\ln(\frac12)=-\ln2$.
J. W. Tanner
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