Chapter 10.5, Problem 50E

To determine

To find: The interval of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(\ln x\right)}^n}$

Answer to Problem 50E

The interval of convergence and the series converges absolutely on $\left(\frac{1}{e},e\right)$

Explanation of Solution

Given information:

The given series is

  ${\displaystyle \sum _{n=0}^{\infty }{\left(\ln x\right)}^n}$

Formula used:

  ${\displaystyle \sum _{n=0}^{\infty }\left|a_n\right|}={{\displaystyle \sum _{n=0}^{\infty }\left|\ln x\right|}}^n$

Calculation:

The series of absolute values is

  ${\displaystyle \sum _{n=0}^{\infty }\left|a_n\right|}={{\displaystyle \sum _{n=0}^{\infty }\left|\ln x\right|}}^n$

This is a geometric series with $r=\left|\ln x\right|$

So, the series converges absolutely for $\left|\ln x\right|<1$ or $e^{-1}<x<e^t$ and diverges for $\left|\ln x\right|\ge 1$

Hence,

(a) Interval of convergence: $\left(\frac{1}{e},e\right)$

(b) Series converges absolutely on $\left(\frac{1}{e},e\right)$

(c) None.

Conclusion:

The interval of convergence and the series converges absolutely on $\left(\frac{1}{e},e\right)$