Finding Alternative Representations for the Convergent Series Ʃ ((-1)^(i-1))/i

[venom192]

Homework Statement

I basically need to know if the series Ʃ ((-1)^(i-1))/i can be represented in other ways (e.g. a Taylor series, but I doubt it). I know it converges to ln2, but I need to know if there's a series like x^2, x^4, ... or something like it that I can represent the series with.

Homework Equations

Ʃ ((-1)^(i-1))/i for i=1 to q, where q is a finite, but very large number

The Attempt at a Solution

My calc textbook doesn't examine that series for anything besides its convergent nature, and I can't find any online resources for it. Any help would be appreciated.

 

[clamtrox]

I'm not sure if I understand what you are asking... But have you tried expanding ln(1+x) as a Taylor series?

 

[venom192]

I'm not sure if I understand what you are asking... But have you tried expanding ln(1+x) as a Taylor series?

Thanks for the reply. This question specifically pertains to the series I asked about - I was just using the Taylor series as a possibility of an alternate representation of the series. After looking at my old calc textbook, I see that I was way off base suggesting a Taylor series.

I'll rephrase this - Is it possible to rewrite the series Ʃ(-1^i)/i as another type of series that contains a quadratic term?

 

[clamtrox]

Quadratic in what? You can of course develop the series at different points. If you solve for the series of ln (3/2+x) then you'll end up with series containing powers of 1/2.