## Is there any equivalent of Taylor/Maclaurin series of $ln(1+x)$ for $|x| > 1$

I happened to come across Taylor series and Maclaurin series recently, but everywhere I read about the expansion for $$ln(1+x)$$, it was stated that the approximation is valid for $$-1 < x < 1$$.

I understand that the bounds for $$x$$ are because the series doesn’t converge for $$|x|>1$$, but is there any equivalent of this series for the value of $$|x|$$ as greater than $$1$$?

Please note that I am not asking if we can compute for $$|x|>1$$ or not, as that can be done by computing for $$\frac 1x$$, which will then lie between $$-1$$ and $$1$$.

Also, I’m quite new to all this, so new that today was the day I read the name ‘Maclaurin’ for the first time. So any answers understandable with high school mathematics are highly appreciated.

Thanks!

## how to prove the inequality $-x\ln(x) – \ln(1-x)> \ln(x)\ln(1-x))$

Can someone please help me in proving the inequality: $$-x\ln(x) – \ln(1-x)> \ln(x)\ln(1-x))$$