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In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence {\displaystyle } defines a series S that is denoted
The nth partial sum Sn is the sum of the first n terms of the sequence; that is,
A series is convergent if the sequence {\displaystyle } of its partial sums tends to a limit; that means that, when adding one a k {\displaystyle a_{k}} after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number ℓ {\displaystyle \ell } such that for every arbitrarily small positive number ε {\displaystyle \varepsilon } , there is a integer N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} ,
If the series is convergent, the number ℓ {\displaystyle \ell } is called the sum of the series.