Week 11: Understanding a very strange behavior

Jun 20, 2022

Hello everyone and welcome back to my blog!

This week I am pleased to have made significant progress on partially explaining a phenomenon that was part of the motivation to work on this topic in the first place.

At the start of the project I had seen (and coded) an example of the Euler-Maclaurin series being applied to the function f(x)=1/x^2. The result of this is a divergent series being equated to a finite value, but when certain values for the formula are chosen, the series appears to converge extremely quickly to this value, but as many more terms are added, the divergent behavior of the series takes over as expected. Proving this series diverges is not super difficult (given the asymptotic behavior of Bernoulli numbers, which appear in the Euler-Maclaurin series) and given the derivation of the formula, why it yields an equation with a divergent sum on one side and a finite value on the other can also be seen (it’s because of switching the order of summation which is not necessarily allowed for infinite sums). What’s perplexing is that this series appears to converge to the value it is assigned in the first place.

This week I was messing around with some graphs on desmos and was looking at the graph of partial sums of the series 0!+1!x+2!x^2+3!x^3+… and the graph of that series’s Borel sum (integral from 0 to infinity of e^-t/(1-xt)dt) and noticed that while the series diverges, if the sum of the first say 10 terms is shown, for a sliver of very small values of x this series appears to be almost exactly equal to its Borel sum. Again, in this case as more and more terms of the series are added, that sliver will shrink until it’s gone entirely because the series is known to diverge. Again, it’s perplexing that that sliver exists in the first place, but I may have a partial explanation which could probably be applied to the original example.

The sum 0!+1!x+2!x^2+3!x^3+…+10!x^10 has the property that if n≤10, then the nth derivative evaluated at 0 is n!^2, while it n>10, then it’s 0. The function (integral from 0 to infinity of e^-t/(1-xt)dt) has the property that for all n, the nth derivative evaluated at 0 is n!^2. The fact that these 2 functions have the same first few derivates at 0 could explain why for values of x very close to 0 the functions are almost exactly the same while it takes a value of x further away from 0 before the different behaviors of the functions (resulting from their different higher order derivates) “kicks in.” I believe that this explanation is hand wavy and incomplete, but may be a good first step for explaining this bizarre behavior.

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