# Basel problem

The , 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 $s=2$ (http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2)

## References

• 1 Ed Sandifer, “Euler’s Solution of the Basel Problem - The Longer Story”. Danbury, Connecticut: Western Connecticut State University (2003)
Title Basel problem BaselProblem 2013-03-22 18:05:22 2013-03-22 18:05:22 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 11A25