Question

Prove that the sum of all natural numbers is -1/12.

Answer 1

This is not true. The infinite series


$\displaystyle\sum_{n=1}^\infty n$

converges if and only if the sequence of its partial sums converges. The $k$-th partial sum is

$\displaystyle\sum_{n=1}^kn=\frac{k(k+1)}2$

but clearly this diverges as $k$ gets arbitrarily large.