FP3 Stuck on quick question

Probably a stupid question, but why is the range of values of x for which the expansion is valid -1/2 < x < 1/2?

http://prntscr.com/dknt6h

Thanks

Do you know that the expansion of $\ln(1+x)$ is valid for $-1 < x \leq 1$? If not: the proof is likely out of your grasp at the moment, but it should be a given result.

Any, then the expansion of $\ln(1+ 2x)$ is valid for $-1 < 2x \leq 1$ and the expansion of $\ln(1-2x)$ valid for $-1 < -2x \leq 1$.

Since you're taking the product of those two series, you need the interval of validity (known as the radius of convergence) to satisfy both of the above inequalities, i.e $-\frac{1}{2} < x \leq \frac{1}{2}$ and $-\frac{1}{2} \leq x < \frac{1}{2}$. From which you get the required result.

Edit: Looking at it now it's easy to understand, not sure why I was confused, haha. Thanks

Since you're taking the product of those two series, you need the interval of validity (known as the radius of convergence) to satisfy both of the above inequalities,

It's the sum of the series, no?