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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 \(n<\infty \).