Interesting Limit (25)

Euler had earlier shown that $\displaystyle \frac {\sin x}{x}=\prod_{n=1}^{\infty} \left(1-\frac {x^2}{n^2\pi^2}\right)$ Setting $x=it$ gives $\displaystyle \frac {\sin it}{it}=\prod_{n=1}^{\infty} \left(1+\frac {t^2}{n^2\pi^2}\right)$

Now using that $\displaystyle \sin x=\frac {e^{ix}-e^{-ix}}{2i}$ and setting $t=\pi$ we get $\displaystyle \frac {e^{\pi}-e^{-\pi}}{2\pi}=\prod_{n=1}^{\infty} \left(1+\frac {1}{n^2}\right)=\xi$ Where $\xi$ is our required limit.

set this limit to L = (1+x²/1²) (1+x²/2²) (1+x²/3²)...

with x = 1

choose y so that x = iy/pi

then L = (1-y²/pi²) (1-y²/2²pi²)...

Euler showed this to be equal to sin(y)/y, which gives L = sin(y)/y

L = (exp(iy) - exp(-iy))/(2iy)

y = x pi/i, with x = 1

L = (exp(pi) - exp(-pi))/(2pi)