Is $\sum_{n=1}^{\infty} {\frac{(-1)^n}{x+2^n}} $ in $(-2,\infty)$ uniformly convergent?

Question

Is $\sum_{n=1}^{\infty} {\frac{(-1)^n}{x+2^n}} $ in $(-2,\infty)$ uniformly convergent?

I started by checking if it is pointwise convergent, because if it wasn't then especially it is not uniformly conveergent. But by Leibnitz, it is convergent.

But I got stuck on proving that it is uniformly convergent. Any help would be appreciated!

Answer 1

Hint: Use Weierstrass M-test with $$ M_n = \frac{1}{2^n-2}.$$

Answer 2

Also a hint: Set $f(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n}}{x+2^{n}}$. Now consider sequences $x_{k}=-2+\frac{1}{2^{k}}$ and $y_{k}=-2+\frac{1}{2^{k+1}}$. Obviously $|x_{k}-y_{k}|\to 0$ when $k\to\infty$. Now see what happens with $|f(x_{k})-f(y_{k})|=|x_{k}-y_{k}||\sum_{n=1}^{\infty}\frac{(-1)^{n}}{(x_{k}+2^{n})(y_{k}+2^{n})}|$. More precisely, consider the first term in summation...