infinite series

English

Noun

infinite series (plural infinite series)

1. (mathematical analysis) Any expression that represents the addition a countably infinite number of ordered summands, often explicitly indexed by the positive or nonnegative integers.

   In 1734, Leonhard Euler solved the Basel problem by summing the infinite series ${\displaystyle \textstyle {\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}}$.

Usage notes

The sum of an infinite series is formally defined as the limit, if it exists, of the sequence of partial sums of the first n elements as n increases to infinity (i.e., becomes arbitrarily large). If the limit exists, the series is said to be convergent; otherwise it is divergent.

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