Answer
$$
1+ \frac{ 1}{2\sqrt {2}}+\frac{ 1}{3\sqrt {3}}+\frac{ 1}{4\sqrt {4}}+\frac{ 1}{5\sqrt {5}}...
$$
The given series converse.
Work Step by Step
$$
1+ \frac{ 1}{2\sqrt {2}}+\frac{ 1}{3\sqrt {3}}+\frac{ 1}{4\sqrt {4}}+\frac{ 1}{5\sqrt {5}}...
$$
this series can be written as
$$
\begin{aligned}
1+ \frac{ 1}{2\sqrt {2}}+\frac{ 1}{3\sqrt {3}}+\frac{ 1}{4\sqrt {4}}+\frac{ 1}{5\sqrt {5}}... &=
\sum_{n=1}^{\infty} \frac{1}{n\sqrt {n}}\\
&=\sum_{n=1}^{\infty} \frac{1}{n^{\frac{3}{2}}}
\end{aligned}
$$
Apply p-series we obtain $p=\frac{3}{2 }\gt 1$.
So, the given series is converse.