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Showing posts from August, 2017

### Various representations for famous mathematical constants

In this unusual post, much like in the older post, The beauty of Infinity, we're listing the most famous mathematical constants as representations of infinite seriesinfinite products and limits.

The list of constants considered in this post, are:
Apéry's constant: \zeta(3)πmathematical constante mathematical constantEuler-Mascheroni constant: γCatalan's constant: \G or \beta(2)Natural logarithm of 2: \log(2) Most of this representations have a very fast convergence. # Representations for Apéry's constant \zeta(3) = 5/2 \sum_{n=1}^(\infty) ((-1)^(n - 1) (n!)^2)/(n^3 (2 n)!)

\zeta(3) = \sum_{n=1}^(\infty) ((30 n - 11) (n!)^4)/(4 n^3 (2 n - 1) ((2 n)!)^2)

 \zeta(3) = -π^2 / 3 \sum_{n=0}^(\infty) (2^(-2 n) (2 n + 5) ζ(2 n))/((2 n + 1) (2 n + 2) (2 n + 3))

\zeta(3) = -(4 π^2)/7 \sum_{n=0}^(\infty) (2^(-2 n) ζ(2 n))/((2 n + 1) (2 n + 2))

\zeta(3) = -(4 π^2)/7 [log(27/16) + \sum_{n=0}^(\infty) (2^(-2 n) (ζ(2 n) - 1))/((2 n + 1) (2 n + 2))]

`\zeta(3) = (…