infinite series

English

Noun

infinite series (plural infinite series)

(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.

Translations

infinite series

Danish: uendelig række
Romanian: serie infinită f
Russian: бесконе́чный ряд m (beskonéčnyj rjad)
Serbo-Croatian: beskonačni red m
Tagalog: awangganing dalayray