Basel and Beyond: An Incomplete History of a Famous Sum (Mathematics Colloquium)
March 27, 2013
Swart Hall 217, 4:30 - 5:30 PM
“If anyone finds and communicates to us that which thus far has eluded our efforts, great will be our gratitude."
With these words, published in 1689, Jacob Bernoulli brought to the attention of European mathematicians a problem first posed by Pietro Mengoli in 1644. The problem was to find an exact value for the infinite sum of reciprocals of squares: 1 + 1/4 + 1/9 + 1/16 + ...
The problem came to be known as the Basel Problem, after the Swiss university town where Bernoulli lived and worked. Fittingly, the problem was solved by Basel's finest mathematician, Leonard Euler, in 1735. In his long and productive career, Euler provided two separate proofs, as well as two efficient ways to calculate the value of the sum.
The Basel Problem continues to intrigue mathematicians. Dozens of proofs have been given, including two published within the last year. We will discuss Euler's first proof, as well as a few more recent proofs. We will also explore some of the many extensions and variations.