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Leonhard Euler (1707 – 1783) ranks among history’s greatest mathematicians. In this talk given Sept. 17, 2015, after a brief introduction to Euler's life and work, Prof. William Dunham presents in full detail two of his great theorems that are ingenious and not widely known.
The first dates from 1737 when Euler investigated the behavior of the sum of the reciprocals of the primes, i.e., 1/2+1/3+1/5+1/7+1/11+⋅⋅⋅. In today’s parlance, he raised the question of whether this infinite series converges or diverges. Needless to say, Euler answered the question correctly, and we’ll see how.
Then, we look at his 1755 evaluation of the sum of the reciprocals of the squares. He had twice before presented arguments for finding 1+1/4+1/9+1/16+1/25+⋅⋅⋅, but this particular attack had a special flair, for it used l’Hospital’s rule not once, not twice, but thrice!
William Dunham is the Truman Koehler Professor of Mathematics at Muhlenberg College.