One of friend gave me a question today to solve which is as follows
$$\sum_{r=1}^{\infty} \sqrt {\frac {r}{r^4+r^2+1}}$$
In spite of much efforts I couldn't solve it and so I asked him to check whether the question was correct or was it this one
$$\sum_{r=1}^{\infty} \frac {r}{r^4+r^2+1}$$
I thought the question would be this one because the terms inside the root can be telescoped in absence of root. And indeed I was right. The question was as I expected the latter one.
But even after that I thought about whether the first question containing the square root could also be solved or not. For just checking out the convergence of the sequence I tried the ratio test but it wasn't quite helpful. Then I tried using the integral test and indeed $$\lim_{t\to \infty} \int_{1}^{t} \sqrt{\frac {x}{x^4+x^2+1}} dx$$
this integral converges to $1.80984$ according to Wolfy. Upon lot of efforts too I am not an able to solve the first summation (with the square root terms) . Can someone please lend me some help over this problem.
We have $$\sqrt{\frac{r}{r^4+r^2+1}}\approx \frac{2}{\sqrt{r-1/2}}-\frac{2}{\sqrt{r+1/2}},$$ so by creative telescoping the value of the series is not too far from $\frac{1+2\sqrt{2}}{\sqrt{3}}$, and this approach can be improved in order to obtain more accurate approximations. However I won't bet on a simple closed form for the given series. Numerical methods produce $\approx 2.12534074896$, which not by chance is pretty close to $$ \int_{1/2}^{+\infty}\sqrt{\frac{x}{x^4+x^2+1}}\,dx\approx 2.12, $$ an incomplete elliptic integral.