Consider a power series
$$
\sum\_{n=0}^{\infty}a\_nz^n
$$
where $a\_n$ and $z$ are complex numbers. There is radius $R$ of convergence. Let us assume that is a positive real number. It is well known that for $|z|<R$ the series converges absolutely; for $|z|>R$ it does not converge.


On the other hand, when $|z|=R$, the series can have very different behaviors. This has been discussed in many posts, e.g. 
[Behaviour of power series on their circle of convergence](https://mathoverflow.net/questions/49395/behaviour-of-power-series-on-their-circle-of-convergence) , [Seeking a Geometric Proof of a Generalized Alternating Series' Convergence](https://mathoverflow.net/questions/109582/seeking-a-geometric-proof-of-a-generalized-alternating-series-convergence) .


I am looking for some relatively easy explicit examples of power series with funny behavior at the boundary. The only I know is
$$
\sum\_n\frac{1}{n}z^n
$$
or small variations, e.g. replacing $z$ with $z^k$.


thanks



Power series with funny behavior at the boundary
The series $$f(z) = \sum\_{n=1}^\infty \dfrac{z^{2^n}}{n}$$
converges almost everywhere on the unit circle by Carleson's theorem (it is the Fourier series of an $L^2$ function). However, it diverges on a dense set, including all the $2^k$'th roots of unity: 
in fact at each of those points the real parts of the partial sums $S\_k = \sum\_{n=1}^k z^{2^n}/n$ are unbounded above. 


Now for each positive integer
$N$, the set $U\_N$ of points $z$ such that $\text{Re}(S\_k) > N$ for some $k$
is open and dense in the unit circle. The intersection $G$ of the $U\_N$ is a dense 
$G\_\delta$ by the Baire category theorem, and the series diverges at every point of $G$.


Putting these facts together, we find that the set of points of the unit circle where the series diverges is negligible in the sense of Lebesgue measure but the set where it converges is negligible in the sense of Baire category. I'd call that funny...


