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I have read about Taylor series and Maclaurian series and in which we can express any function in a series having terms $x$ and its power and usually we first assume that it can really be written in a infinite polynomial series and then by derivating and set $x=0$ we calculate the coefficients now I come to know that these function can not be written in polynomial series over the entire region of $x$ value like to express $\dfrac{1}{(1-x^2)}$ we must have $|x|<1$. Similarly many functions can be written in series in different region ,

(1) my question is why I can't express it for the entire region ??

(2) And can they be expressed in some other form excluding that region?

(3) And what is the region to express $\dfrac{1}{(1+x)^2}$ ?

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    $\begingroup$ You cannot express ANY function in such terms, there are some conditions that must be satisfied. And the region in which you can express a function with the series depends on the en.wikipedia.org/wiki/Radius_of_convergence; it does not hold everywhere. $\endgroup$ – Giuseppe Bargagnati Jun 20 '17 at 9:55
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    $\begingroup$ In complex analysis it's shown that the radius of convergence equals the distance to the nearest pole. The poles of $1/(1-x^2)$ are $\pm i$ so the radius of convergence around $0$ is $\min(|(+i) - 0|, |(-i) - 0|) = 1$. $\endgroup$ – md2perpe Jun 20 '17 at 10:29
  • $\begingroup$ my recommendation: take a book of analysis that had chapters about power series, convergence and analyticity and read it to understand in depth (1) and (3). A brief introduction to Fourier series will help also for the part (2) of your question. $\endgroup$ – Masacroso Jun 20 '17 at 11:45

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