If the series is given by $$S=\sum_{i=1}^\infty a_i$$, and the partial sum to n terms defined by $$S_n = \sum_{i=1}^n a_i$$, then S is said to be convergent if for any $$\epsilon >0$$ N can be found such that $$|S_k-S_N|<\epsilon$$ $$\forall k>N$$. A Taylor polynomial is a Taylor's series truncated after the first n terms, where $$n<\infty$$.