Continuity on boundary of convergence power series

Question

I’m stuck trying to prove the following: Given $f(z) = \log(2+z^2)$, I consider its power series representation around $z=0$, which is $i(z) = \log(2)+\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n2^n}z^{2n}$. I have computed its radius of convergence, $R=\sqrt{2}$, I know the equality between function and series on the open disk $D(0,\sqrt{2})$ and that the series converges $\forall z \in \partial D$ if $z \neq \pm \sqrt{2}i$, but I’m struggling to analyse the continuity of the series $i(z)$ restricted to that set on the boundary. I’m aware that the series will be continuous on $|z|<\sqrt{2}$ because there we have uniform convergence, but not on the circunference. Setting $z=\sqrt{2}e^{i\theta}$, I’d want to prove continuity of $i(\theta) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}e^{2i\theta n}$ or relate it with a continuous function on that set, like $\log(1+z)$. But I don’t know how to follow from this. Any help?