when is a series said to be convergent
Convergent series A series is convergent if the sequence of its partial sums   { S 1  ,   S 2  ,   S 3  , … }  {\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   ℓ  {\displaystyle \ell }  such that for any arbitrarily small positive number   ε  {\displaystyle \varepsilon } , there is a (sufficiently large) integer   N  {\displaystyle N}  such that for all   n ≥   N  {\displaystyle n\geq \ N} ,
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