Complexified

$x^{\frac{2}{n}}+6=5x^{\frac{1}{n}}$ take $x=y^{n}$ $\Rightarrow y^2+6=5y$ $\Rightarrow y^2-5y+6=0$ $\Rightarrow y= \frac{5\pm1}{2}$ $\Rightarrow y = 2 or 3$ $\Rightarrow x=2^n or 3^n$ By taking reciprocals and pluging in $\theta(n)$ , we get $\theta(n)=\frac{1}{6^n}$ $\Rightarrow \sum_{n=1}^{\infty}\theta(n)=\sum_{n=1}^{\infty}\frac{1}{6^n}$ As $\sum_{n=1}^{\infty}\frac{1}{s^n}=\frac{1}{s-1}$ $\Rightarrow \sum_{n=1}^{\infty}\theta(n)=\frac{1}{5}$