I am working on the following power series. (Cant copy images yet)
I have applied the ratio test on the series ignoring the $-ln(2)$, and have reduced it down to
$\frac{xn }{2(n+1)}$
When i take the limit of this i end up with $x/2$. I know the interval of convergence is abs$(x) < 2$.
Do i need to include the $-ln(2)$ in this part, or have i done something wrong in the calculation, as $x/2$ doesnt seem to be right. Also, does the $-ln(2)$ affect the actual values of the interval and radius of convergence.
I also concluded it was absolutely convergent for all values, and conditionally convergent for none. Is this correct?
Thanks.
Your power series convergent for $|x| < 2$, this is correct. You do not need to include the term $-\ln(2)$ in your considerations, because it doesn't affect the convergence or the radius of convergence. But it seems to me as if you didn't fully understand how to use the ratio test....your limit is correct, it is $\frac{|x|}{2}$ and this value has to be $< 1$, hence $|x| < 2$ as stated above. So don't forget the absolute value with $x$. Moreover, the power series is absolutely convergent for $|x| < 2$ and divergent for $|x| > 2$. The convergence behaviour at the boundary values $x = \pm 2$ needs to be evaluted separately. To check this, insert these two values in your power series and use appropriate convergence criteria to see if you get convergent/divergent series (Hint: [alternating] harmonic series!).