Find the Taylor series for f(x) = ln(x) centered at x = 1. Whats the radius of convergence?

Question

Please help me out with this question.

I'm having trouble finding the f^n(c) for ln(x).

i'm having trouble finding the radius convergence

Answer 1

To find the Taylor series for ln(x) centered at x = 1.

Remember that a power series is equal to the following:

(1/1-x) = 1 + x + x^2 + x^3 .... (x^n)

If you substitute an x-1 into the above equation
you get
(1/x) = 1 + (x-1) + (x-1)^2 + (x-1)^3 .... (x-1)^n

If you integrate you will get the following series:

ln(x)

Remember that if you can derive other series from integration
and differentiation

Edit:

Use the ratio test to find the radius of convergence.

In order to apply the ratio test to this question you
should have the polynomial approximation of the natural lograrithm expressed in a power series.

Then take the limit of the power series ratio.

Remember that if the ratio test is inconclusive
if the limit of the expression as it approaches infinity is equal to one.

Sorry if I have confused you. Hope this helps.