- #1

ellynx

- 5

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**1. Alternating power series question on convergence interval.**

I'm wrestling a bit with an alternating power series, the teacher has the convergence interval to be

[tex]x = <-2,2][/tex] and I don't agree.

## Homework Equations

Without further adue, here is the alternating power series in question:

[tex]\sum^{\infty}_{n=1}(-1)^{n-1}\frac{x^{n}}{n*2^{n}}[/tex]

Ratio test says;

[tex]\lim_{n=\infty}|\frac{x^{n+1}}{(n+1)*2^{n+1}}\frac{n*2^{n}}{x^{n}}|[/tex]

and end at the fruity limes below

[tex]\frac{|x|}{2}\lim_{n=\infty}|\frac{n}{(n+1)}|[/tex]

now I understand that, [tex]\lim_{n=\infty}|\frac{n}{(n+1)}| \approx 1[/tex]

and in order to satisfy the criteria, ratio < 1, the limits for x must lie in <-2,2>.

Inserting limits for x in the original series i got:

[tex]x=2: \sum^{\infty}_{n=1}(-1)^{n-1}\frac{1}{n}, converges[/tex]

[tex]x=-2: \sum^{\infty}_{n=1}(-1)^{2n-1}\frac{1}{n}, diverges[/tex]

## The Attempt at a Solution

This far I agree with the limits stated by the teacher, but something bugs me, I computed the series with many different values for x>2 in maple and the result was a finite real number every time. For positive x values the series seems to alternate and converge, even though it brakes the ratio < 1. Is it possible that this series is conditionally convergent, or am I on the wrong mind track?

Cheers.