As a counterexample -- while looking for two divergent sequences whose sum or product converges we have the obvious choice, (1, -1, 1, -1, . . . ) with (-1, 1, -1, 1 . . . ) but is there another cool example of this?
As a counterexample -- while looking for two divergent sequences whose sum or product converges we have the obvious choice, (1, -1, 1, -1, . . . ) with (-1, 1, -1, 1 . . . ) but is there another cool example of this?
EDIT: Oooops... you asked for sequences and I give you a series...(But it can work as well)
$\displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}}$ is converges.
But, $\displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}}\cdot\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n}} $ isn't.
Edit: disregard this. The left forum sidebar protruded over the formula for $\displaystyle S_2$ and obscured the minus.
Isn't $\displaystyle S_1+S_2$ the harmonic series? Then it does not converge.