Interval of convergence of $\sum_{n=1}^\infty x^{\ln(n)}$.

Question

What is the interval of convergence of this series or for what values of $x$ does it converge? $$\sum_{n=1}^\infty x^{\ln(n)}$$

I tried the ratio and root test but they were inconclusive, any help would be appreciated.

Answer 1

If you allow me to use Wolfram alpha

All those tests are inconclusive

And it is converging if, $$\ln |x| +1\lt 0$$

We can write $$x^{\ln n}$$ $$e^{\ln n ×\ln x}$$ $$n^{\ln x}$$ This is the famous reimann zeta function, $$\sum_{n=1}^{\infty} n^{-s}$$ Where $s=-\ln x$

$$\sum_{n=1}^{\infty} \frac{1}{n^{-\ln x}}$$

Here is the proof

http://planetmath.org/PTest

Answer 2

The series is only defined for $x > 0$. Under that assumption, $x^{\ln n} = n^{\ln x}$, so by the $p$-series test, $\sum_{n=1}^\infty n^{\ln x}$ converges if and only if $\ln x < -1$, or $x \in (0, e^{-1})$.