Is this complex function harmonic? Let us consider the following convergente series in the set $0<x<1$ and all real $y$:
$$h(x+iy)=∑_{n=2}^{∞}(-1)ⁿ⁻¹((n^{2x-1}-1)/n^{x})n^{iy}$$
My question is: Is this complex function harmonic? 

Look at the terms of the series. Ignoring the $(-1)^n$ for the moment, we have
$$\frac{n^{2x-1}-1}{n^x}n^{iy} = n^{x-1}n^{iy} - n^{-x}n^{iy} = n^{z-1} - n^{-\overline{z}}.$$
The first term is holomorphic, and hence harmonic. The second term is antiholomorphic, and hence harmonic. Thus the difference of the two terms is harmonic.
So every term in the series
$$h(x+iy) = \sum_{n=2}^\infty (-1)^n \frac{n^{2x-1}-1}{n^x}n^{iy}$$
is harmonic, and since the series is locally uniformly convergent, it follows that $h$ is harmonic.
