Bounds on a complex sequence

Question

Let $y>0$ and $0<x<1$ be fixed. Is the sequence $$\frac{(n+1)^{x+iy}-n^{x+iy}}{(n+1)^{x}-n^{x}},\,\,n\geq 1,$$ bounded?

Answer 1

Let $a_n=\frac{(n+1)^{x+iy}-n^{x+iy}}{(n+1)^x-n^x}=\frac{(1+\frac{1}{n})^x(n+1)^{iy}-n^{iy}}{(1+\frac{1}{n})^x-1}\approx \frac{n^{iy}\frac{x+iy}{n}}{\frac{x}{n}}=n^{iy}(1+\frac{iy}{x})$

Since $|n^{iy}|=1$, the sequence is bounded.

The approximation uses $(n+1)^{iy}=n^{iy}(1+\frac{1}{n})^{iy}\approx n^{iy}(1+\frac{iy}{n})$ and $(1+\frac{1}{n})^x\approx 1+\frac{x}{n}$ for large $n$.