Week 8: Borel Summation

Jun 02, 2022

Hello everyone and welcome back to my blog!

This week I learned about Borel Summation and made substantial progress on tweaking a number of proofs that I had wanted to fix. Like Cesaro summation, Borel Summation assigns values to divergent sums by having the property that if a series is convergent, the Borel sum of the series will be equal to that convergent sum, but many sums that are divergent have a Borel sum.

Borel summation is applied to a power series and works by considering the power series with each coefficient divided by n!, and then applying a transformation (which I have dubbed to Gamma transform, though I don’t believe this is lingo used by anyone else) which for polynomials will multiply each coefficients by n!, thus for convergent power series, this just gives the original power series, but for divergent power series, this may create a new function that is convergent. For example, the series 0!-1!+2!-3!+… is (very) divergent, but with Borel summation, this can be assigned the value (really we are not assigning it a value, but rather associating a different quantity with this series rather than a typical sum) the integral from 0 to infinity of the function e^(-x)/(1+x), which is about .596.

Because whether a series is Borel summable relies only on the convergence of the series where each coefficient is divided by n!, Borel summation can be applied to many more series than Cesaro summation, but like Cesaro summation, only alternating sums. This is because Borel summation really just switches the order of summation which can only change the behavior of alternating series. I wish to include a table demonstrating this in my final presentation.

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