The Evolution of Euler's Summability in Mathematical Arena

Abstract

Euler was the first mathematician who studied infinite series in depth, especially focusing on divergent series. He asserted that divergent series must have some definite value. He became successful in this direction by finding a system to sum such series. In these days, Euler's summation methods are popular with some modifications in different names like Cesaro, Abel, Hölder, Nörlund, Hausdorff matrices, Borel summation, and so on. In this paper, I discuss some of Euler's work on summation methods of infinite series associated with the Euler-Maclaurin summation formula. We also examine Euler's effort on the solution to the Basel problem and his devotion to Grandi's series with some examples.