What is the mistake in this attempt to prove \sum {\frac{x^n}{n}} = -ln(1-x)?

[talolard]

Homework Statement

I am trying to show that $$\sum{ \frac{x^n}{n}} = -ln(1-x)$$
But I am doing something wrong and I can't find my mistake.
Please find my mistake and let me know what it is.
Thanks

The Attempt at a Solution

set $$f(x)=\sum {\frac{x^n}{n}}$$
then $$f'(x)= \sum {x^n-1}$$
so $$xf'(x)=\sum{x^n} = \frac {1}{1-x}$$
which means that $$f'(x)=\frac {1}{x} - \frac{1}{1-x}$$
integrtang we get $$f(x)=ln|x|-ln|1-x|$$
which is bad

 

[timthereaper]

The derivative of the function is:

f'(x)=Sum[x^(n-1),{n,1,Infinity}]

which is the same as:

f'(x) = Sum[x^(n),{n,0,Infinity}] = 1/(1-x)

Integrating:

Sum[x^(n+1)/(n+1),{n,0,Infinity}] = - ln(1-x)

which is the same as:

Sum[x^(n)/n,{n,1,Infinity}] = -ln(1-x).

The trick lies in repositioning the starting index.