Find the region of convergence and absolute convergence of the series. sum_{n=1}^inftyfrac{ln^n x}{n}

Question

Find the region of convergence and absolute convergence of the series.

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

Answer 1

Given the series, $\sum_{n = 1}^{\infty} \frac{\ln^{n} x}{n}$
Here, $| a_{n} | = | \frac{(\ln x)^{n}}{n} |$
To find the region of convergence and absolute convergence of the given series.
By ratio test, if $lim_{n \to \infty} \frac{a_{n + 1}}{a_{n}} < 1$ , the series converges.
Therefore, the absolute convergence of the series implies,
$lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | = lim_{n \to \infty} | \frac{(\ln x)^{n + 1}}{n + 1} \times \frac{n}{(\ln x)^{n}} |$
$= lim_{n \to \infty} | \frac{n \ln x}{n + 1} |$
$= lim_{n \to \infty} | \frac{\ln x}{1 + \frac{1}{n}} |$
$= | \ln x |$
So, the series absolutely converges if, $| \ln x | < 1$
Therefore, the region of convergence is, $| \ln x | < 1$