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What could be an intuitive explanation for $\displaystyle \sum_{k=1}^{\infty}{\frac1{k\,2^k}} = \log 2$ ?

$\displaystyle \sum_{k=1}^{\infty}{\frac{1}{2^k}} = 1$ has a simple intuitive explanation with taking a unit distance and adding the successive halfs consecutively, is there a similar explanation for $\ln 2$?

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    Oddly enough someone else just asked this question yesterday, before deleting it. Anyway, try the Taylor expansion of $-\log(1/2)$.2011-08-05
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    In order for their to be an "intuitive" explanation, you'd have to start with an intuitive way to understand $\ln 2$ in the first place.2011-08-05
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    @John: This isn't a coincidence: http://meta.math.stackexchange.com/questions/2729/2011-08-05
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    @John I am the culprit, though it was not yesterday but the day before I think. I have posted an answer.2011-08-06
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    log is the inverse function to taking powers. You sum, more or less, over powers, which is, more or less, like integrating over them, and we know this is related to values of the antiderivative.2011-08-06
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    There's also the related formula $\ln2=\sum_{k=1}^\infty\frac{(-1)^{k+1}}k=1-\frac12+\frac13-\dotsb$, which one can prove _without calculus_ (but starting with the assumption that $e^x\ge x+1$ for all $x$). I don't know if a similar proof exists for your series, though.2015-11-10

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