“Mom, I have the feeling I’m the Ramanujan or Paul of the 21st century”.
Today Simon learned that it was Euler who first came up with the idea that the infinite sum 1 + 2 + 3 + 4 +… converges to -1/12. Simon explained to me the other day that there are several ways of looking at an infinite sum. One way is looking at its partial sums and summing those up. Another way is averaging partial sums and see what their average converges to (or what the average of their averages converges to). “That’s called Cesaro Summation and it’s good for closely related sums like 1 – 1 + 1 – 1 + 1… but not for 1 + 2 + 3 + 4 +…”, Simon explained. “Then there is Ramanujan Summation – a Calculus way of looking at infinite sums using derivatives and gamma, etc. That is the only way 1 + 2 + 3 + 4 +… converges to -1/12. All possible infinite sums converge if you use Ramanujan Summation.”
“Simon, you don’t trust Ramanujan Summation, do you?” I asked.
“No. Only an infinitely small section of infinite sums converge using the standard method. Converging means it settles down. That’s what we call a fixed point. If an infinite sum doesn’t converge, it can either explode to infinity or it can have more than one fixed point or do something else weird like that. Sums that are not convergent are called divergent.”
“And Ramanujan had none of those, no divergent sums?”
“Yeah, he really made it to the extreme! It’s an infinitely large extreme. All infinite sums Ramanujan-converge.”
Ramanujan’s Taxicab Numbers are the smallest numbers that can be written as a sum of two numbers to the power of n in two different ways. We only know such numbers if n is 1, 2, 3 and 4. We don’t know the smallest number that can be written as a sum of two fifth powers in two different ways, or any other powers larger than four. The name “Taxicab Numbers” comes from the story about Ramanujan’s visit to Cambridge, when he got picked up by a cab with 1729 on the number plate.
“Every possible integer is one of Ramanujan’s personal friends”, Simon tells me after he sketches the Taxicab numbers at a beach cafe. “Which is a bit uncomfortable because you have infinitely many friends then”.