Challenge Level

Investigate the following sequence of fraction sums:

\begin{eqnarray} \frac{1}{2} &+&
\frac{2}{1} = \\ \frac{2}{3} &+& \frac{3}{2} = \\
\frac{3}{4} &+& \frac{4}{3} = \\ \frac{4}{5} &+&
\frac{5}{4} = \end{eqnarray}

What would you get if you continued this sequence for ever?What do you think will happen if you add the squares of these
fractions, that is:

\begin{eqnarray} \left(\frac{1}{2}\right)^2
&+& \left(\frac{2}{1}\right)^2 = \\
\left(\frac{2}{3}\right)^2 &+& \left(\frac{3}{2}\right)^2 =
\end{eqnarray}

and so on?