Let's be clear about terms.
A sequence is a set of terms; more properly it is a function defined on the natural numbers; call the terms an.
The sequence can converge to a limit L, if for every ε>0, there exists N such that |an - L| < ε whenever n > N.
By a series is usually meant the sequence of partial sums, sn = ∑ ak from k =0 to n.
Convergence to a "sum" S means that for every ε>0 there exists N such that |sn - S| < ε whenever n > N.
You should notice that what I have defined are, in fact, 2 limits. The only other basic piece of information I will supply in this answer is that if L ≠ 0, then S cannot exist.
If I have not adequately answered your question, please send me a message via the "comment".
