Short answers, pulled from the story.
Zeno of Elea constructed his paradoxes about infinite sums in the 5th century BCE. These paradoxes argued that motion was an illusion because an infinite sequence of steps could never be completed. The resolution required the development of the mathematical concept of the limit over two millennia later.
Archimedes used infinite series to calculate precise geometric areas in the 3rd century BCE. He employed the method of exhaustion to sum an infinite geometric series and determine the area under a curved arc. His work proved that the area of a parabolic segment was exactly four-thirds the area of the largest inscribed triangle.
Neelakanta Somayaji wrote the Tantrasangraha around the year 1500. This Sanskrit text described series expansions for sine, cosine, and inverse tangent functions. Jyesthadeva provided detailed proofs in the Malayalam text Yuktibhasa, completing the work around the 1st of January 1530.
Carl Friedrich Gauss published his memoir on hypergeometric series in 1812. This publication established simpler criteria for convergence and raised the question of when a series actually had a sum. Augustin-Louis Cauchy followed this in 1821 by insisting on strict tests of convergence.
Bernhard Riemann proved in the 19th century that any conditionally convergent series of real numbers can be rearranged to converge to any arbitrary real number. He also showed that such series can be rearranged to diverge entirely. This result, known as the Riemann series theorem, demonstrated that the sum of a series depends critically on the order in which they are added.
Cesàro summation assigns a value of 1/2 to Grandi's series. Grandi's series is 1 - 1 + 1 - 1 + ... and oscillates between 0 and 1 with no limit. Techniques such as Cesàro summation, Abel summation, and Borel summation provide systematic ways to assign generalized sums to series that fail to converge.