Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

Given an infinite sequence ${\displaystyle \left(a_{1},\ a_{2},\ a_{3},\dots \right)}$, the nth partial sum ${\displaystyle S_{n}}$ is the sum of the first n terms of the sequence, that is,

A series is convergent if the sequence of its partial sums ${\displaystyle \left\{S_{1},\ S_{2},\ S_{3},\dots \right\}}$ tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number ${\displaystyle \ell }$ such that for any arbitrarily small positive number ${\displaystyle \varepsilon }$, there is a (sufficiently large) integer ${\displaystyle N}$ such that for all ${\displaystyle n\geq \ N}$,

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