Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$ where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime numbers of the form $4m-1$ have a $-$ sign, prime numbers of the form $4m+1$ a $+$ sign, but composite numbers have that sign which agrees with the account of the multiplication from the primes.
He proved it by combining $$\frac{\pi}{4}=\prod_{p\,\text{prime},p\gt 2}\frac{1}{1-(-1)^{\frac{p-1}{2}}p^{-1}}$$ (which is a special case of the Euler product of the Dirichlet beta function) with $$\frac{1}{(1-az)(1-bz)(1-cz)\cdots}=1+Az+Bz^2+\cdots$$ where $$A=\text{sum of the individual terms},$$ $$B=\text{sum of the two factors at a time},$$ etc. with the same factors not excluded, and set $z=1$.
But, there is one problem with Euler's proof. He is multiplying infinitely many geometric series and collecting like powers – treating it like polynomial multiplication, this surely requires some justification; can we prove that the factors in $$\frac{1}{(1-az)(1-bz)(1-cz)\cdots}$$ (where $a,b,c,\ldots$ are the signed reciprocal primes) can be arbitrarily rearranged?
Euler is not very rigorous and I can't figure it out myself – how to save Euler's proof?
Added: GH from MO has already provided an answer on MO: https://mathoverflow.net/questions/459675/eulers-proof-of-frac-pi6-1-frac12-frac13-frac14-frac15