I have a slightly daft question concerning Taylor series.

Say if I want to approximate a function about a point $a$, and I am using a finite Taylor series of order $m$, then this series will be a "decent" approximation of my function only in a small neighborhood around $a$.

However, when we are approximating functions by infinite Taylor series (which have a certain ratio of convergence $R$), we still have to define a center $a$. What I am confused about, is whether in this case the function will be "decently" approximated once again in a very small neighborhood around the center $a$ or in the whole interval, $|x-a|< R$? Is this what distinguishes the two series? Is the term approximation applicable here (for infinite series)? (Or is the function exactly equal in this interval).

  • $\begingroup$ Everywhere in your interval of converges Taylor series converges to original function. $\endgroup$ – EzWin Dec 18 '16 at 12:23
  • $\begingroup$ There is no finite series, I mean you want to say Taylor polynomial instead of finite series. $\endgroup$ – Masacroso Dec 18 '16 at 12:24
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    $\begingroup$ @Msacroso link. But yes, I forgot about functions like $f(x) = e^{-1 / x^2}$ if $x\neq0$, else 0. But i was talking about analytic functions, tbh. $\endgroup$ – EzWin Dec 18 '16 at 12:44

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