Q:

History of infinite series

When was $\sum$ introduced as the notation for a sum and who was the first person to solve a infinite sum other than 0+0+0+...?

A:

According to Florian Cajori's History of Mathematical Notations Vol II paragraph 438, the symbol $\displaystyle \sum$ for summation is attributed to Leonard Euler. In Euler's own words, "summam indicabimus signo $\sum$". Cajori cites Euler's Institutiones calculi differentialis (1755) in his sourcing.
As for the first known infinite sum, I do see some online sources attributing the first known infinite sum to Archimedes. In particular, on the site A history of the calculus, the authors John J. O'Connor and Edmund Robertson cite Archimedes' showing around 225 BC that "the area of a segment of a parabola is $\frac{4}{3}$ the area of a triangle with the same base and vertex and $\frac{2}{3}$ of the area of the circumscribed parallelogram," as the first known example of an infinite sum. Archimedes lived from 287 to 212 BC.
Personal opinion has it that while this may be the first known infinite summation, there may well have been one before that not known to us today. 

