Question

MEDAL Find a power series representation for the function and determine the radius of convergence, R. f(x) = ln(12 − x)?

Answer 1

After integrating the power series, I'll have

C − 1/12 sum n=0 to infinity of x^n + 1/ ????

Answer 2

So we can start here,\[\large\rm \ln(12-x)=\int\limits\frac{-1}{12-x}dx\]Factoring a 1/12 out gives us this,\[\large\rm =-\frac{1}{12}\int\limits\frac{1}{1-\frac{x}{12}}dx\]rewriting this as geometric series in summation,\[\large\rm =-\frac{1}{12}\int\limits\sum_{n=0}^{\infty}\left(\frac{x}{12}\right)^n~dx\]Passing the integration procedure into the sum,\[\large\rm =-\frac{1}{12}\sum_{n=0}^{\infty}\int\limits \left(\frac{x}{12}\right)^n~dx\]Integrating,\[\large\rm =-\frac{1}{12}\sum_{n=0}^{\infty}\frac{12}{n+1}\left(\frac{x}{12}\right)^{n+1}\]

Answer 3

There are some of the steps.
Any confusion?
Having trouble with the integration?

Answer 4

Oh ya ya, I should include the C,
that's important,\[\large\rm =C-\frac{1}{12}\sum_{n=0}^{\infty}\frac{12}{n+1}\left(\frac{x}{12}\right)^{n+1}\]