Proving particular functional series converges

Question

I am interested in proving that the following series converges uniformly: $$\sum_{n=1}^{\infty}\frac{(-1)^{(n+1)}}{x+(n-1)}$$ for all $x\in (0,\infty)$. I anticipate that the key will be in proving the Cauchy criterion for the sequence of functions given by $$f_n(x)=\frac{(-1)^{(n+1)}}{x+(n-1)}$$ i.e. that given $\epsilon>0$ there exists some $N$ s.t. for all $m>n\geq N$, $$\Big|\sum_{k=m+1}^{n}f_k\Big|<\epsilon$$ for all $x\in (0,\infty)$. I have tried to use the triangle inequality to show that the function is less than $(m-n)/p$ where $p$ is a number s.t. $\frac{1}{p}<\epsilon$, but I have not had much success as ultimately $m-n$ will depend on $N$, and I need some similar invariant which will determine $N$ in order to reduce the inequality down to $\epsilon$.

Answer 1

HINT:

Apply Dirichlet's Test for Uniform Convergence. Note that

$$\left|\sum_{n=1}^N(-1)^{n-1}\right|\le 1$$

Now, for what values of $x$ does $\frac{1}{x+n-1}$ converge uniformly?