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\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_408_1433_296_315} The diagram shows the curve with equation \(y = x ^ { - 2 }\) for \(2 \leqslant x \leqslant N\) together with a set of ( \(N - 2\) ) rectangles of unit width.

1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } } > \frac { 3 } { 2 } - \frac { 1 } { N } + \frac { 1 } { N ^ { 2 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_2718_35_141_2012}
2. Use a similar method to find, in terms of \(N\), an upper bound for \(\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } }\).
3. Deduce lower and upper bounds for \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\).