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# Basel problem

The Basel problem, first posed by Pietro Mengoli in 1644, asks for a finite formula for the infinite sum

$\sum_{{i=1}}^{{\infty}}\frac{1}{i^{2}}.$ |

Though Mengoli verified the Wallis formulae for $\pi$, it did not occur to him that $\pi$ was also involved in the solution of this problem. Jakob Bernoulli also tried in vain to solve this problem. Even an approximate decimal value eluded contemporary mathematicians: an answer accurate to just five decimal places requires iterating up to at least $i=112000$, which without the aid of a computer was wholly impractical in Mengoli’s day. The problem was finally solved in 1741, when, after almost a decade of work, Leonhard Euler conclusively proved that

$\frac{\pi^{2}}{6}=\zeta(2)=\sum_{{i=1}}^{{\infty}}\frac{1}{i^{2}}.$ |

The value, 1.6449340668482264365… could then be computed to almost as many decimal places as were known of $\pi$. See value of the Riemann zeta function at $s=2$

# References

- 1 Ed Sandifer, “Euler’s Solution of the Basel Problem - The Longer Story”. Danbury, Connecticut: Western Connecticut State University (2003)

## Mathematics Subject Classification

11A25*no label found*

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