The function is represented as a power series .
Find the first few coefficients in the power series.
Find the radius of convergence R of the series.
To the first question: Recall that the power series expansion for is the series (this is derived by taking the derivative of the well know power series for )
So we have that:
therefore, we have
use that to find the first few coefficients
For the second question:
use the root test (or if you prefer, the ratio test, either one works nicely in this case) to find the radius of convergence. do you need help with that part?
Ok, we wish to find the radius of convergence for the series
Since you didn't specify what test you wanted me to use, I will pick one myself. I choose the root test, it's less typing than the ration (why do i always do that?! type "ration" when i mean to type "ratio"!) test in this case.
We have
we have convergence if
Thus the radius of convergence is