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Basel Problem

Why

1

Problem

Consider the sequence $(a_n)_{n \in \N }$ defined by

\[ a_n = \frac{1}{n^2}. \]

Does $\lim_{N \to \infty} \sum_{n = 1}^{N} a_n$ exist? If so, what is the limit? These questions are known as the Basel problem.

Solution

\[ \lim_{N \to \infty} \sum_{n = 1}^{N} s_n = \frac{\pi ^2}{6}. \]

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