A Disputable Problem

If this sum converge to A then every subsequence would converge to same limit A, but in the sequence $a_n = \sum_{i=1}^n (-1)^{i+1}$ associated to the sum above you can take the subsequence with even terms, $a_2 , a_4 , a_6, ...$ and $\lim_{n \to \infty} a_{2n}$ = (1 - 1) + (1 - 1) + (1 -1) + ... = 0, and you can take the subsequence $a_1 , a_3 , a_5 , ...$ and $\lim_{n \to \infty} a_{2n + 1}$ = 1 + ( -1 +1) + (- 1 +1) + (- 1 +1) + (- 1 + 1) +... = 1 $\Rightarrow$ the sum above doesn't converge.

Note: Euler believed wrongly that this sum precisely would converge to 0.5, it was necessary to aximomatize Aritmethic and Maths