# Doubt in Euler's proof Basel problem

In the proof of the Basel problem Euler reaches a step where expansion of $(\sin x)/x$ is $[1-(x^2/\pi^2)][1-(x^2/4\pi^2)][1-(x^2/9\pi^2)]\cdots$ And from that he got to- $1- x^2\{1/\pi^2 + (1/4\pi^2) + (1/9\pi^2) +\cdots\}+ x^4(\cdots)+\cdots$. I don't understand how this happens?

• It comes from the second derivative at $x=0$. To make this rigorous you need some complex analysis to show $\pi \cot (\pi z) = \lim_{N \to \infty} \sum_{n=-N}^N \frac{1}{z+n}$. – reuns Oct 14 '17 at 8:41
• Well, what do you get as the coefficient of $x^2$ when you multiply everything out? – Gerry Myerson Oct 14 '17 at 8:46
• But how do we multiply "everything" when we have to multiply infinite "things"? – user167920 Oct 14 '17 at 8:48
• Well x^2 will be preserved only if the term containing x^2 is multiplied by 1. – user167920 Oct 14 '17 at 8:50
• @GerryMyerson Yes I do believe so. Thanks for your help – user167920 Oct 15 '17 at 12:47