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The Basel Problem

by

Professor Paul Friedman
Union College

November 2, 2015
5:00 pm
Bailey Hall 207

Refreshments will be served in Bailey Hall 204 4:45pm


Abstract:

In 1644, Pietro Mengoli posed the following problem: Find the numerical value of the sum of the reciprocals of the squares, that is, \begin{align*} \text{evaluate } 1+\frac 14+\frac 19+\frac 1{16}+\cdots = \sum_{k=1}^\infty \frac 1{k^2}. \end{align*} At the time, mathematicians were able to show that this sum was indeed finite (unlike the harmonic series, 1 + 1/2 + 1/3 + 1/4 + ··· ). However, it was not until 90 years later that Euler was able to evaluate this sum. In this talk, we will present (one of) Euler's computations that this sum is, remarkably, π2/6. In essence, his work relies on a study of the sine function, writing it both as an infinite polynomial (a Taylor series, as in Math 110, 113, and BC Calculus) and as an infinite product. The result follows somewhat naturally from comparing theses two considerations. After that, time permitting, we will discuss another proof of the same result that uses integration techniques from Math 117.


For additional information, send e-mail to math@union.edu or call (518) 388-6246.
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