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Geometric Mean

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Tolaso J Kos
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Geometric Mean

#1

Post by Tolaso J Kos » Tue Aug 09, 2016 8:06 pm

Find the geometric mean , with respect to the usual measure on the interval, of all the real numbers in the range $(0, 1]$.
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Riemann
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Re: Geometric Mean

#2

Post by Riemann » Sat Oct 22, 2016 9:50 am

Nice one. The geometric mean with respect to the usual measure on the interval is actually defined as:

$$\mathcal{GM} = e^{\displaystyle \int_{0}^{1} \ln x \, dx} = e^{-1} = \frac{1}{e}$$

and this is our answer.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$
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