Q:

Finding convergence of the next function: $f(x)=\sum_{n=1}^\infty\frac{(\ln n)^3}{n}x^n$

How can i find whether the next function converges: $f(x)=\sum_{n=1}^\infty\frac{(\ln n)^3}{n}x^n$?
I thought about this question for quite a while, What's the trick?

A:

The ratio test here will work nicely:
Recall that we apply the ratio test to the the entire summand:
$$a_n=\left({(\ln n)^3\over n}|x|^n\right)$$
and doing so here will give you the result $\;1\cdot |x|.\;$ So the series will converge $\forall x: |x| \lt 1$, and will diverge $\forall x:|x| > 1$.
What happens at $x = +1, x = -1\,$ remains to be determined, so be sure to test for convergence/divergence at $\;x = 1,\;$ and also at $\; x = -1.\;$

