## anonymous 3 years ago If a serie is convergent does it mean the serie can be rewritten as a taylor polynomial but it can't be rewritten if it's divergent? If not what's the meaning of convergence I don't think I get the textbook approch?

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 $$n<\infty$$.