Why is:$$\sum_{n=1}^\infty \left(\frac{(-1)^{n-1}(x-1)^n}{n}\right) \neq \sum_{n=0}^\infty \left(\frac{(-1)^{n}(x-1)^{n+1}}{n+1}\right)$$
This question arose when I tried to get the series representation for ln(x) [centered @ x=1] using geometric series method. I couldn't get it to match. I originally thought maybe it had something to do with the fact that I had: $$\frac{1}{1-(1-x)} = \int\sum_{n=0}^\infty \left({(1-x)^n}\right)dx $$Vs:
$$\frac{1}{1-(-(x-1)} = \int\sum_{n=0}^\infty \left({(-1)^{n}(x-1)^n}\right)dx $$ But both of the above 2 are the same correct? So I just can't wrap my brain around why the very 1st two summations aren't the same. Don't Geometric series require the index to start at n=0? $$$$I feel like there's some basic math principle I have forgotten.. Thanks for the help!