$X(n)$ and $Y(n)$ divergent doesn't imply $X(n)+Y(n)$ divergent. Please, give me an example where $X(n)$ and $Y(n)$ are both divergent series, but $(X(n) + Y(n))$ converges.

Try $x_n = n, y_n = -n$. Then both $x_n,y_n$ clearly diverge, but $x_n+y_n = 0$ clearly converges.
Or try $x_n = n, y_n = \frac{1}{n}-n$ if you want something less trivial. Again both $x_n,y_n$ diverge, but $x_n+y_n = \frac{1}{n}$  converges.
