# 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+...?

• $\sum$ stands for summation be it an infinite series or not. – Omar Nagib Sep 24 '15 at 18:04
• The sum of the geometric progression like 1+1/2+1/4+...is known since the time immemorial, so the question who "was the first" does not make sense. – Alexandre Eremenko Sep 24 '15 at 19:24
• The people who did this did not use the modern Greek sigma, but why should one care what letter is used??? – Alexandre Eremenko Sep 24 '15 at 19:25

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.