"Proof" that $1-1+1-1+\cdots=\frac{1}{2}$ and related conclusion that $\zeta(2)=\frac{\pi^2}{6}.$

If by $\sum$ you mean what is usually meant, then $$ \sum_{k=0}^{\infty} e^{ki\vartheta} $$ diverges, and the first formula and the rest of the proof is invalid.

This serie doesn't converge in the usual sense (partial sum converging towards a limit), as you can extract sub-sequences that converge towards 1 or 0. But there are alternative definition of summation, like Cesaro or Abel that will make this converge.

Euler spent a lot of time trying to decide wether or not it would make sense to say that this converges.

Wikipedia articles:

1-2+3

Cesaro

Edit: For the record, this is the Dirac comb. It makes sense to admit the convergence to 1/2 if you're thinking of it as a Fourier transform.

You discovered a very interesting result. Its validity depends on your definition of the summation.

In the usual sense the series is divergent and doesn't have a sum. So it's invalid.

However, your equation is valid if you define the summation to be the Cesaro summation, in which case the limit of the arithmetic mean of the first partial sums of the series is used.

This type of results is widely used in physics, for example, in string theory.