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Leonhard Euler (1707 – 1783) ranks among history’s greatest mathematicians. In this talk, after a brief introduction to his life and work, I present in full detail two of his great theorems that are ingenious and (I hope) not widely known.

The first dates from 1737 when Euler investigated the behavior of the sum of the reciprocals of the primes – i.e., $1/2 + 1/3 + 1/5 + 1/7 + 1/11 + \cdots$. 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 $1 + 1/4 + 1/9 + 1/16 + 1/25 + \cdots$, but this particular attack had a special flair, for it used l’Hospital’s rule not once, not twice, but thrice!!

These two results, taken from Euler’s vast opus, should make clear why he was aptly called “The Master of Us All.”