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## Geometric Mean

- Tolaso J Kos
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**Articles:**2**Posts:**854**Joined:**Sat Nov 07, 2015 6:12 pm**Location:**Larisa-
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### Geometric Mean

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

**Imagination is much more important than knowledge.**

### Re: Geometric Mean

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.

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

and this is our answer.

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