I've been looking at some pretty cool proofs of $\zeta(2)=\frac{\pi^2}{6}.$ recently, and the one proof that was the easiest to understand for me was how Euler originally presented it, by finding and manipulating the infinite series for $\frac{\sin(x)}{x}$. Through the the wild and mostly ridiculous thought process that leads me to most of my cooler mathematical experiments, I thought: "What if I made an infinite sum out of that infinite series?" Infinitiception. So, I ended up researching the sum in question: $$\sum_{x=-\infty}^{\infty}\left(\dfrac{\sin(x)}{x}\right)\;$$
and it turns out it is well known as $\pi$! This was awesome. I love pi, and while the connection here is quite obvious, it is still fun to see it pop up everywhere. But then, through some playing around, I found this:
$$\sum_{x=-\infty}^{\infty}\left(\dfrac{\sin(x)}{x}\right)\;= \sum_{x=-\infty}^{\infty}\left(\dfrac{\sin(x)}{x}\right)^{2}=\pi\;$$ This simply doesn't seem possible to me. How can a series equal itself squared? $\sin^2 \neq \sin$, and obviously $x^2 \neq x$, so how can this be true? Is it even true? I would appreciate an explanation with as little calculus as possible, unless it is needed or more elegant with calculus. Thanks!
There's no paradox here. It's quite possible for the sum of a bunch of numbers to equal the sum of the squares of those numbers. This happens even in finite sums: $$ \frac13+\frac13+\frac43=\bigg(\frac13\bigg)^2+\bigg(\frac13\bigg)^2+\bigg(\frac43\bigg)^2 $$ $$ -\frac13+\frac12+c=\bigg({-}\frac13\bigg)^2+\bigg(\frac12\bigg)^2+c^2 \text{ for either } c = \frac{3\pm\sqrt2}6. $$