Infinity, as an informal concept, is associated with endless repetition. Common expressions like "and so on" and "etcetera" and, occasionally, the ellipses "..." reflect a concept that mathematics attempts to make rigorous. On this page I shall collect a few appealing formulas whose meaning I hope will be intuitively clear even without formal justification. I believe this could be a good way to illustrate one of the uses of infinity in mathematics.
Geometric series: \(\sum_{n\ge 0}2^{-n}\)
\(2^0 + 2^{-1} + 2^{-2} + 2^{-3} + \ldots = 2.\)
Telescoping series: \(\sum_{n\ge 1}\frac{1}{n(n+1)}\)
\(\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \frac{1}{4\cdot 5} + \ldots = 1.\)
James Gregory's (or Leibniz) series
\(\frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots = \frac{\pi}{4}\).
Euler's series: \(\sum_{n\ge 1} n^{-2}\)
\(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi ^2}{6}.\)
Euler's series: \(\sum_{n\ge 1} (2n-1)^{-2}\)
\(\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \cdots = \frac{\pi ^2}{8}.\)
Euler's alternating series: \(\sum_{n\ge 1}(-1)^{n+1}n^{-2}\)
\(1 - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots = \frac{\pi ^2}{12}.\)
Euler's alternating series: \(\sum_{n\ge 1}(-1)^{n+1}(2n-1)^{-3}\)
\(1 - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + \frac{1}{9^3} - \frac{1}{11^3} + \cdots = \frac{\pi ^2}{32}.\)
Alternating Harmonic Series:
\(\frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots = ln(2)\).
Nilakantha (15th century) I:
\(\frac{1}{1^5+4\cdot 1} - \frac{1}{3^5+4\cdot 3} + \frac{1}{5^5+4\cdot 5} - \frac{1}{7^5+4\cdot 7} + \ldots = \frac{\pi}{16}.\)
Nilakantha (15th century) II:
\(3+\frac{4}{3^3-3} - \frac{4}{5^3-5} + \frac{4}{7^3-7} - \frac{4}{9^3-9} + \ldots = \pi.\)
