Find the intervals of convergence of f(x), f'(x) f''(x) and for the function Check for convergence at endpoints of each inverval.
I've been having a lot of trouble with problem and I'd really appreciate any help I can get.
For f(x), apply the ratio test and find the limit as . Then check the endpoints as well. It converges at x=2.
By the ratio test:
The ratio test gives cionvergence for and divergence for
Solving the first inequality we get .
At x=1, the series is , which is a divergent p-series.
At x=3, it is convergent and converges to -ln(2).
So, the convergence set is
Check it out. Make sure I did not err somewhere, then run a test on the derivatives of the series.
You can use the alternating series test to show convergence.
But if x=3, you get
This is a series I knew converges. This is the series associated with
the "Euler's constant". Which shows
As for the antiderivative of your series. Integrating wrt x, wouldn't it be