# Powerseries of a natural logarithm

1. Oct 11, 2011

### center o bass

1. The problem statement, all variables and given/known data
Show that

$$\frac{1}{x} \ln \frac{x+1}{x-1} = \sum_0^\infty \frac{2x^{2n}}{2n+1}$$.

2. The attempt at a solution
I tried to use the relation

$$\frac{1}{1+x} = \frac{d}{dx}\ln (1+x)$$

and expand as a geometric series, but this did not lead anywhere since I then ended up with an alternating series.

Anyone got any ideas of where to start?

2. Oct 11, 2011

### Hootenanny

Staff Emeritus
HINT:

$$\log\left(\frac{a}{b}\right) = \log a - \log b$$

3. Oct 11, 2011

### center o bass

Yes, so I could subtract the two resulting series... and I see now that it will actually lead trough. Ofcourse I knew about the property, but I did not see how it would get rid of the 'alternation'. However I do see that now because of the cancellation of terms.

Thank you!