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So as we know, the power series by Euler of the exponential function converges for every x. That means that for all x and for all epsilon, we can find an N so that the series converges. Is there a way to compute this N for each x?

In other words, there exists a function $f: \mathbb{R} \to \mathbb{N}, x \mapsto N$. Do I have any knowledge about this $f$?

anon
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  • https://math.stackexchange.com/questions/4482530/approximate-ex-with-bounded-size-sliding-window-over-its-taylor-series/4482625#4482625 – Claude Leibovici Jul 02 '22 at 07:59
  • Are you interested specifically in the least $N$ such that the series is within $\epsilon$ of $\exp(x)$ for $n \ge N$, or any such $N$? In the latter case, the usual proof of convergence (ratio test...) is actually constructive, and you can extract some concrete values at each stage. It does give you a very conservative estimate, though. – Izaak van Dongen Jul 03 '22 at 23:11

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