Solution
= Solution
The <series> $\sum a_n$ converges when its partial sums tend to a finite real <limit>. Put $m=n+1$. Then
$$
a_n=\frac{2\sqrt m}{\sqrt{m-1}+\sqrt{m+1}}-1>0
$$
by strict concavity of the square root. Taylor expansion at $m$ gives
$$
\sqrt{m-1}+\sqrt{m+1}=2\sqrt m\left(1-\frac1{8m^2}+O(m^{-4})\right),
$$
so $m^2a_n\to1/8$. <Limit> comparison with $\sum n^{-2}$ proves convergence.