\[
\displaylines{\text{Is }\sum a_{n}\text{ convergent if}\\ 1)\text{ }a_{n}=\sqrt{n+1}-\sqrt{n}\\ 2)\text{ }a_{n}=\frac{\sqrt{n+1}-\sqrt{n}}{n}\\ 3)\text{ }a_{n}=\left(\sqrt[n]{n}-1\right)^{n}\\ 4)\text{ }a_{n}=\frac{1}{1+z^{n}}\text{ for }z\in\mathbb{C}}
\]