Power Series Convergence Question.

Question

I have a power series shown below.(Cannot link pictures yet).

Power series

I am unsure how to deal with the constant of -ln2. If that term is removed, i have found the series to converge when absolute value of x is less than 2. Do i simply add on the constant to that inequality?. Also, how does it affect the absolute and conditional convergence of the series?

Thanks

Answer 1

The constant term makes no difference when determining whether or not the series converges. In fact, the first 1,000,000,000 elements make no difference when all you want to know is whether the series converges or diverges.

So, in this case, your originally computed radius of convergence for the series is correct. If you are claiming that the interval of convergence is specified by $\lvert x\rvert<2$, be careful -- this series actually converges when $x=-2$ (it becomes the alternating harmonic series), so that the interval of convergence is actually $-2\leq x<2$.