Here we will try to apply the standard tests to determine the convergence of the series
We treat x as parameter and sum up with respect to k. What
tests can be used? Comparison seems out, since for k large we cannot
really ignore anything. Actually, the expression
This would suggest that this series converges for all values of x, which is obviously wrong, see below.
So comparison is out, how about the popular twins, Ratio and Root? We start with the Root test.
We see that the given series converges absolutely on the interval
We also know that for
One can also show this directly. First we rewrite the terms of our series like this:
|ak| = |x|k⋅|1 − x|.
Now we see that for
It remains to individually check on the points