I have an exercise based on the exercise 8 of page 41 in Complex Analysis of Ahlfors. In that exercise they ask for the values of x in which the following series converges:
$$ \sum_{n=0}^\infty{(\frac{x}{1+x})^n} $$
It is clear that the series converges for $x >-1/2 $
However, I am trying to prove the uniform convergence of the series in this interval. My guess is that I have to find a correct bound and use the Weierstrass M-test, but I do not have any idea on which boound would be succesful.
Thanks!
$x> -1/2\implies 1+x> 1/2\implies \frac{1}{1+x}< 2\implies -\frac{x}{1+x}< 1\implies |\frac{x}{1+x}|< 1$ , for $x\not= 0$.
Expand the series now..for $x\not=0$.
$$S(x)=\lim_{n\to \infty}S_n(x)= \lim_{n\to \infty} \sum_{k=0}^n{(\frac{x}{1+x})^k} $$
$$=\left(1-\frac{x}{1+x}\right)^{-1}=1+x$$
Thus , $$S(x)=\begin{cases}0 & \text{ if } x=0\\1+x&\text{ if } x\not= 0\end{cases}$$
Since , $S(x)$ is NOT continuous at $x=0$ so, the given series is nNOT uniformly convergent.