Week 6: Cesaro Summation and Tweaking Proofs

Hello everyone and welcome back to my blog!

This week I learned about Cesaro summation. Most of them time, when we talk about the sum of a series, we are referring to its partial sums or infinite sum (which is the limit of its partial sums). The Cesaro sum of a series is a different sum that we can associate to a series, but should not be thought of as the same as the standard “sum of a series.” The Cesaro sum of a series is the limit of the average of a series’s partial sums (there are also higher order Cesaro sums).

Series can either have a sum and have a Cesaro sum (convergent series will always have a Cesaro sum that approaches some value), not have a sum and have a Cesaro sum (divergent series with a Cesaro sum that approaches some value), or not have a sum and not have a Cesaro sum (divergent series where the Cesaro sum does not approach a certain value).

In the first case, it can be proven that the standard sum and Cesaro sum are the same value.

In the second case, sense the ceries is divergent, you cannot say that the sum of the series is the Cesaro sum, but the Cesaro sum shares many properties with ordinary summation (for example, linearity). It may be possible to tweak some “proofs” where divergent sums are incorrectly assigned values and instead correctly state an equivalence between the Cesaro sum of a series and the value that the series was assigned. Unfortunately, this could only work for a limited class of divergent sums, alternating divergent sums (and more specifically alternating divergent sums where the terms behave asymptotically like a polynomial). Divergent sums that are not alternating will never have a convergent Cesaro sum, and sums like: 1-3+9-27+81-… will also not have a convergent Cesaro sum (of any order).

Unfortunately, in the third case, Cesaro summation does no good for us.

I also found this cool quote from Niels Henrik Abel: “The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.” Cesaro summation is just one of the ways that these demonstrations can be tweaked to be correct, and I look forward to exploring other methods.