Answer
Domain : $(1,\infty)$
Work Step by Step
$\Sigma_{1}^{\infty}\frac{1}{n^{x}}=\int_{1}^{\infty}\frac{1}{t^{x}}dt$
when $x=1$ ; $\int_{1}^{\infty}\frac{1}{t^{x}}dt=\int_{1}^{\infty}\frac{1}{t}dt=lnt|_{1}^{\infty}=\infty$
when $0\lt x \lt 1$ ; $\int_{1}^{\infty}\frac{1}{t^{x}}dt=\frac{1}{1-x}[t^{(1-x)}|_{1}^{\infty}=\infty$
when $x \gt 1$ ; $\int_{1}^{\infty}\frac{1}{t^{x}}dt=\frac{1}{1-x}[t^{(1-x)}|_{1}^{\infty}=\frac{1}{1-x}(-1)=\frac{1}{x-1}$
Hence, the given function is defined when $x \gt 1$
Thus, Domain : $(1,\infty)$
