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 Post subject: Geometric MeanPosted: Tue Aug 09, 2016 8:06 pm

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 841
Location: Larisa
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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Imagination is much more important than knowledge.

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 Post subject: Re: Geometric MeanPosted: Sat Oct 22, 2016 9:50 am

Joined: Sat Nov 14, 2015 6:32 am
Posts: 159
Location: Melbourne, Australia
Nice one. The geometric mean with respect to the usual measure on the interval is actually defined as:

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

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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$

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