Limit of Summation of series

$r$th term of a series is $t_r=\frac{r}{1-r^2+r^4}$. Then how do we compute $\lim_{n \to \infty }\sum_{r=1}^n t_r$.

I tried converting the summation into a definite integral so as to use Newton Leibnitz theorem , but was unable to do so. I don't see how to incorporate "$n$" into the general term. Please somebody help. It would be very helpful if someone gives the full solution as I am attempting such a summation - based limit question for the first time.