how to find domains of convergence of a series

Question

Let $f_n(x) = \dfrac{x^n}{n^x}$.

Find domain of convergence of the series $\sum_{n=1}^{\infty}f_n(x)$.

Any help?

Answer 1

Ratio test:

$$\lim_{n\to\infty}\left|\frac{x^{n+1}n^x}{x^n(n+1)^x}\right|<1$$

For $|x|<1$. For $|x|=1$, it diverges by known series.

As a side note, we have the polylogarithm:

$$\sum_{n=1}^\infty\frac{x^n}{n^x}=\operatorname{Li}_x(x)$$

Answer 2

Use some series comparisons for the positive numbers and the Alternating Series Test for negative numbers. $\frac{x^n}{n^x}\leq x^n$ when $0\leq x<1$, so it converges from $0\leq x<1$. At negative values of $x$, you can use the AST. The absolute value of the sequence decreases continuously for $-1