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.

We have $\dfrac{\sqrt{n+2}-\sqrt{n+1}}{\sqrt{n+1}-\sqrt{n}}\dfrac{|x+1|^{n+1}}{|x+1|^n}\to|x+1|$. By the ratio test, the series conveges in $(-2,0)$ and diverges in $(-\infty,-2)\cup(0,\infty)$.
Now, $\sum_{k=1}^n(\sqrt{k+1}-\sqrt{k})=\sqrt{n+1}-1\to\infty$. Thus series diverges for $x=0$.
Can you solve for $x=-2$? 
