Date: Jan 26, 2013 8:52 AM
Author: A N Niel
Subject: Re: Series Convergence
In article <2013012613252172156-bpa2@mecom>, Barry <bpa2@me.com> wrote:

> I have come to a complete dead stop with the following question:

>

> Find all real numbers $x$ such that the series

>

>

> \[

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

> \]

> converges.

>

> I am aware that

>

> \[

> \sum_{n=1}^{\infty}\frac{x^n}{n}=-\log(1-x)

> \]

> and that

> \[

> \sum_{n=1}^{\infty}\frac{1}{n}

> \]

> does not converge.

>

> Any guidance on how to proceed would be much appreciated by this hobby

> student (not on a formal course).

>

If $x>1$ or $x <= -1$ the term does not go to zero ... DIVERGE.

If $-1 < x < 1$, compare to $\sum (-1/n)$ .... DIVERGE.

IF $x=1$, all terms zero ... CONVERGE.