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A N Niel
Posts:
2,240
Registered:
12/7/04
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Re: Series Convergence
Posted:
Jan 26, 2013 8:52 AM
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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.
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