This identity is usually proved by induction, here is the real ‘frontal attack’ from Euler who gave the first convergent sum of (pi^2/6) for Zeta (2).

The Basel Problem is:

$latex displaystyle sum_{k=1}^{infty} frac {1}{k^2} = frac {{pi}^2}{6}$

Euler was 28 years old when he proved that it converged.

The Basel Problem is also called the Riemann Zeta function: ζ(2).

He studied the function sin x which has zeroes,

i.e. sin x= 0 for

$latex x=npi, n = 0,pm1,pm 2, pm 3…$

In other words, we can factor sin x this way:

$latex sin x = x.(1+frac {x}{pi}) .(1-frac {x}{pi}).(1+frac {x}{2pi}). (1-frac {x}{2pi}).(1+frac {x}{3pi}). (1-frac {x}{3pi})…

&s=3$

Note: the right side any factor = 0 when

$latex x=npi, n = 0,pm1,pm 2, pm 3…$

$latex frac {sin x}{x}

= (1-frac {x^2}{1^{2}{{pi}^2}}).

(1-frac {x^2}{2^{2}{{pi}^2}}).

(1-frac {x^2}{3^{2}{{pi}^2}})…

&s=3

$

Note: $latex (1+a).(1-a)= 1 – a^{2}$

From Taylors series,

$latex

sin x = +frac {x}{1!} – frac {x^3}{3!} + frac {x^5}{5!} – frac {x^7}{7!} +…

&s=2$

$latex

frac {sin x}{x} = 1 – frac {x^2}{3!} + frac {x^4}{5!} –…

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