0
$\begingroup$

Find all the real values of $x$ for which $\sum_1^{\infty}(x^n/n)$ converges.

I began with the ratio test to get $nx/(n+1)$ but I'm not sure where to go next. I think I'm supposed to use Leibniz's Theorem at some point?

  • 0
    You should have $x^n$ in the denominator ?2012-11-25
  • 0
    Yes, sorry that was a typo, I'll correct it.2012-11-25

1 Answers 1

3

$$a_n:=\frac{x^n}{n}\Longrightarrow \left|\frac{a_{n+1}}{a_n}\right|=\left|\frac{x^{n+1}}{n+1}\frac{n}{x^n}\right|=|x|\frac{n}{n+1}\xrightarrow [n\to\infty]{}|x|$$

Thus, the series converges for $\,-1

  • 0
    That doesn't really make it any clearer; would the last term not tend to 0 as n tended to infinity? Also, how is it 'easy to see' that it converges for x=-1? How have you used Leibniz's theorem?2012-11-25
  • 0
    Are you familiar with the alternating series test, Mathlete?2012-11-25
  • 0
    Mathlete, when $n=1000000000000$, is $n/(n+1)$ close to zero? or is it close to $1$?2012-11-25
  • 0
    OK I see what @DonAntonio meant now.2012-11-25