$X(n)$ 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.
 A: How about $X_n = -n$ and $Y_n = n$ for $n \geq 1$.
A: 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.
A: Take $y_n = -x_n$ for some divergent $x_n$ since $y_n$ will be divergent too.
A: As others have already pointed out, if $X(n)$ and $Y(n)$ diverge, this need not always mean that $X(n) + Y(n)$ diverges.
A slightly general example is of the following form.
Consider a convergent sequence $c(n) \to c \in \mathbb{R}$. Now pick any divergent sequence $X(n)$. Consider the sequence $Y(n) = -X(n) + c(n)$.
Then we have that if $X(n) \to \infty$, then $Y(n) \to -\infty$ and if $X(n) \to -\infty$, then $Y(n) \to \infty$. Finally, $X(n) + Y(n) \to c \in \mathbb{R}$ and hence it converges.
However, note that if we want $X(n)$ and $Y(n)$ to diverge, but $X(n) + Y(n)$ to converge, then one of them must diverge to $+ \infty$ and the other must diverge to $-\infty$.
