Convergence of series with root Given
$$\sum_{n=1}^\infty {((-1)^n + \alpha^3) (\sqrt{n+1} - \sqrt{n})}$$
find all values of $\alpha$ such that the series converges.
My try:
By multiplying the series with the expression 
$$\frac {\sqrt{n+1} + \sqrt n} {\sqrt{n+1} + \sqrt n}$$
we get
$$\sum_{n=1}^\infty {((-1)^n + \alpha^3) (\sqrt{n+1} - \sqrt{n}) \left(
\frac {\sqrt{n+1} + \sqrt n} {\sqrt{n+1} + \sqrt n}
\right)}$$
$$= 
\sum_{n=1}^\infty {
((-1)^n + \alpha^3) \left(\frac 1{\sqrt{n+1} + \sqrt{n}} \right)
}
$$
Let $$a_n = ((-1)^n + \alpha^3)~,~b_n = \frac 1{\sqrt{n+1} + \sqrt{n}} $$
then, it is easy to show that $b_n$ is monotonously decreasing and
$$\lim_{n \to \infty} b_n = 0$$ 
I know that by Dirichlet rule, if i can find $\alpha$ such that $\sum_{n=1}^\infty a_n$ is bound, than the series will converge.
1.How do i find such an alpha? (if this is the right way...)
2.The Dirichlet rule is not iff. can i use it in this case?

You may write
$$
\begin{align}
\sum_{n=1}^N {((-1)^n + \alpha^3) (\sqrt{n+1} - \sqrt{n})} & =\sum_{n=1}^N {\frac{(-1)^n}{\sqrt{n+1} + \sqrt{n}}+ \alpha^3 \sum_{n=1}^N  (\sqrt{n+1} - \sqrt{n})} \\&= \sum_{n=1}^N {\frac{(-1)^n}{\sqrt{n+1} + \sqrt{n}}+ \alpha^3 (\sqrt{N+1}-1)}
\end{align}
$$
and, as $N \rightarrow \infty$, it is easy to conclude.
