<Dl> <Dd> S n = ∑ k = 1 n a k . (\ displaystyle S_ (n) = \ sum _ (k = 1) ^ (n) a_ (k).) </Dd> </Dl> <Dd> S n = ∑ k = 1 n a k . (\ displaystyle S_ (n) = \ sum _ (k = 1) ^ (n) a_ (k).) </Dd> <P> 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), </P> <Dl> <Dd> S n − l ≤ ε . (\ displaystyle \ left S_ (n) - \ ell \ right \ vert \ leq \ \ varepsilon .) </Dd> </Dl>

When is a series said to be convergent
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