Limits and series

$\large \mathfrak S=\displaystyle \sum_{r=1}^{n}\dfrac{r}{1+r^2+r^4}\\\large= \displaystyle \sum_{r=1}^{n}\dfrac{r}{(r^2+1-r)(r^2+1+r)}$ $\large =\dfrac{1}{2}(\displaystyle \sum_{r=1}^{n}\dfrac{1}{r^2+1-r} - \dfrac{1}{r^2+1+r})\\ (\color{#0C6AC7}{\small{\text{A Telescopic series)}}}$ $\large =\dfrac{1}{2}(1-\dfrac{1}{n^2+n+1})$ $\large=\dfrac{1}{2}(\dfrac{n(n+1)}{(n^2+n+1)})$ $\Large\text{As }n\rightarrow {\infty},~\mathfrak{S} \text{ approaches}\huge\color{#456461}{\boxed{\color{#D61F06}{\boxed{\color{#007fff}{\textbf{0.5}}}}}}$