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?

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mathmate · Answer

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 $.