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Lower Bound

Age 11 to 14 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?