I know that the Taylor Series of(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

f(x)= \frac{1}{1+x^2}

[/tex]

aroundx_{0}= 0

is

[tex] 1 - x^2 + x^4 + ... + (-1)^n x^{2n} + ... [/tex] for |x|<1

But what I want is to construct the Taylor Series of

[tex]

f(x)= \frac{1}{1+x^2}

[/tex]

aroundx. I tried working out the derivatives, but trying to find a general formula for the n_{0}= 1derivative is almost impossible (by the fourth derivative I was already suffocating:P). The thing is, I need this because I want to apply the ratio test (or any other) to find the Radius of Convergence of the series centered around^{th}x. And the reason I want to do this, is because I really want to know if the radius of convergence is zero or not! And the reason I'm interested in this is, because even though_{0}= 1

[tex]

f(x)= \frac{1}{1+x^2}

[/tex]

is indefinitely derivable atx=1, I want to know if it's analytic at this point or not! I want to know if its Taylor series centered around that point has a non-zero radius of convergence, and if it does, if the residue term given by Taylor's Theorem goes to zero as n goes to infinity within the radius of convergence of the series. The reason I picked out this example is because I know that althoughfdoesn't seem to have a singularity atx=1in the real domain, the reason the Taylor expansion offaroundxstops converging at x=1 is because in the complex domain,_{0}= 0x=iand-iare singular points, so I'm hopeful that the Taylor expansion aroundxwill not converge at all in any neighbourhood of_{0}= 1x._{0}= 1

And then I will have found a function that is indefinitely derivable at a point and yet not analytic at that point, which is what I am searching for :) (the only examples of such functions I've ever come across are those whose derivatives at the point of consideration are all zero).

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# Taylor series of 1/(1+x^2) . . . around x=1!

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