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====== Math 445 lecture 10: more ''for'' ====== ===== example 1 ===== Recall this classic formula for $\pi^2/6$ due to Euler: \begin{eqnarray*} \frac{\pi^2}{6} = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \ldots \end{eqnarray*} Previously, we showed how to sum the first $N$ terms of this series with the Matlab one-liner <code matlab> Rewrite this with summation notation \begin{eqnarray*} \frac{\pi^2}{6} = \sum_{n=1}^{\infty} \frac{1}{n^2} \end{eqnarray*} The Nth partial sum $P_N$ of the infinite series is \begin{eqnarray*} P_N = \sum_{n=1}^{N} \frac{1}{n^2} \end{eqnarray*} Then $\pi^2/6 = \lim_{N \rightarrow \infty} P_N$. Now