Thursday, April 12, 2007

Lenny's in the Infinity, Doin' Ur Sums!

A big idea that people started understanding when calculus came along was that you could add up an infinite number of things and get a finite answer, but only if the things you were adding were getting small enough fast enough.

The simplest infinite sum isn't listed above, but it's the idea that if I have a bottle of water, and I drink half the water in the first hour, then half of what's left in the next hour, then half of what's left in the third hour, and continue this pattern, I will never be completely finished. So 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 if I add up an infinite number of fractions following this pattern, but if I only add up any finite part of this, the total will be less than 1.

It was well known before Euler that 1/1 + 1/2 + 1/3 + 1/4 + ..., known as the harmonic series, gets bigger and bigger and if you pick any large number, eventually you can add up a finite part of this list and it will be more than the large number you chose. When this happens, we say an infinite sum is divergent. There was a famous family of Swiss mathematicians, the Bernoullis, who had figured out the the sums of the reciprocals of the squares had to be less than 2, so that means the sum is convergent, but they couldn't figured exactly what number the total would be.

It was a tough problem. They were really good. But when problems get this tough, it's Lenny to the rescue! Yay! He figured out the total of 1/1 + 1/4 + 1/9 + 1/25 + 1/36 + ... is pi squared divided by six.

Lenny also figured out that if we took the prime numbers, (2, 3, 5, 7, 11, 13, 17, 19, ...) and then added up all the reciprocals, that series is divergent. The reciprocals of the primes aren't getting small enough fast enough.

You may notice that Lenny looks kinda squinty in the picture. He lost his vision in one eye relatively early in his life, and was completely blind by the time he was fifty, and he kept working! In this way, he was similar to the great composers Bach, who went blind, and Beethoven, who went deaf.

My Favorite Lenny.

No comments: