Show that the following serie is uniformly convergent in $\mathbb{C}$
$$\sum_{n = 1}^{\infty}\frac{(-1)^{n-1}}{n + |z|}$$
This serie is absolutely convergent in $\mathbb{C}$?
I showed that the series is not absolutely convergent, just consider $z = 0$ and we have the harmonic serie. In addition, using the Leibniz test to alternating series I also showed that the series is pointwise convergent, but I can't show that the serie is uniformly convergent, someone can help me? Thank you
Best Answer
The series converges uniformly for all $z \in \mathbb{C}$ by the Dirichlet test which states that $\sum a_n(z)b_n(z)$ converges uniformly if the partial sums $\sum_{k=1}^n a_k(z)$ are uniformly bounded in modulus and $b_n(z) \downarrow 0$ as $n \to \infty$ monotonically and uniformly.
Here we have for all $n \in \mathbb{N}$ and $z \in \mathbb{z}$,
$$a_k(z) = (-1)^{k-1} \implies \left|\sum_{k=1}^n a_k(z) \right| \leqslant 1$$
We also have
$$b_n(z) = \frac{1}{n + |z|} \leqslant \frac{1}{n} \to 0,$$
where it is clear that the convergence to $0$ is uniform and monotone since $b_{n+1}(z) \leqslant b_n(z)$.