- #1
BobbyBear
- 162
- 1
I know that the Taylor Series of
[tex]
f(x)= \frac{1}{1+x^2}
[/tex]
around x0 = 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]
around x0 = 1. I tried working out the derivatives, but trying to find a general formula for the nth derivative 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 x0 = 1. 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
[tex]
f(x)= \frac{1}{1+x^2}
[/tex]
is indefinitely derivable at x=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 although f doesn't seem to have a singularity at x=1 in the real domain, the reason the Taylor expansion of f around x0 = 0 stops converging at x=1 is because in the complex domain, x=i and -i are singular points, so I'm hopeful that the Taylor expansion around x0 = 1 will not converge at all in any neighbourhood of x0 = 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).
[tex]
f(x)= \frac{1}{1+x^2}
[/tex]
around x0 = 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]
around x0 = 1. I tried working out the derivatives, but trying to find a general formula for the nth derivative 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 x0 = 1. 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
[tex]
f(x)= \frac{1}{1+x^2}
[/tex]
is indefinitely derivable at x=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 although f doesn't seem to have a singularity at x=1 in the real domain, the reason the Taylor expansion of f around x0 = 0 stops converging at x=1 is because in the complex domain, x=i and -i are singular points, so I'm hopeful that the Taylor expansion around x0 = 1 will not converge at all in any neighbourhood of x0 = 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).