convergence of series: $ \sum_{n=1}^\infty(\sqrt{n+1}-\sqrt{n})\cdot(x+1)^n $ I would like to prove the convergence of series: $$ \sum_{n=1}^\infty(\sqrt{n+1}-\sqrt{n})\cdot(x+1)^n $$ for x $\in \mathbb{R}$. I am a bit lost on this one. I guess I would be interested in 


*

*$x<-1$

*$x = -1$

*$x > -1$
Any help would be greatly appreciated.

Hint: you want to use the Root test so you want to calculate
$$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \dfrac{\sqrt{n+2}-\sqrt{n+1}}{\sqrt{n+1}-\sqrt{n}}\dfrac{|x+1|^{n+1}}{|x+1|^n} = |x+1|$$
and find the values for which the limit is $< 1$.
So the series converges
$$\forall x | |x+1| < 1\\
\forall -1 < x + 1 < 1\\
\forall -2 < x < 0$$
