I have the question to express $\displaystyle f(x)= \frac{2}{4+x} $ as a series and determine when it converges.
This seems to work out pretty easily to $ \sum_{k=0}^\infty 2(-x-3)^k $ , and this seems to work for values $ -4
However, the answer in the book is $\displaystyle \sum_{k=0}^\infty (-1)^k\frac{1}{2^{2k+1}}x^k $, with $ -4
I keep going over the proof that $ \sum_{k=0}^\infty ar^k $ converges to $\displaystyle \frac{a}{1-r} $ if $ |r| < 1 $ but I can't figure out what I'm missing.