# Summability Question

Today I was reading Cesaro Summability and Abel summability. I found that there exists a series which is Cesaro summable but do not converge in conventional way (the usual way...). Again there exists a series which is Abel summable but not Cesaro summable.

My Question: Is this type of summability has an end. I mean is there any rule, say $P$... after that you can not impose any other rule $Q$ such that we will not have the following: There exists a series which is $Q$ summable but not $P$ summable.

Btw is there any other summability which dominates Abel Summability?

• Generally no. Once you impose a rule on summability someone else can generalise or change one of your hypotheses to generate a new rule of summability. An excellent reference is Hardy's Divergent Series (archive.org will let you view it here: archive.org/details/DivergentSeries but if you can get hold of the AMS reissue it's a lovely book) which will give you just about every useful way to sum a series. – postmortes Aug 24 '17 at 19:06
• – Simply Beautiful Art Aug 24 '17 at 20:13
• thanks @postmortes – MAN-MADE Aug 25 '17 at 2:07