<p>Let the $r$-th term, $t_r$ of a series is given by $t_r = \frac{r}{1 + r^2 + r^4}$. The value of $\lim_{n \to \infty} \sum_{r=1}^{n} t_r$ is</p>
Step-by-Step Solution
Key Concept: Decompose the general term using partial fractions to create a telescoping series where consecutive terms cancel.
<p>Use partial fractions: $\frac{r}{1 + r^2 + r^4} = \frac{1}{2}\left(\frac{1}{r^2 - r + 1} - \frac{1}{r^2 + r + 1}\right)$.</p><p>The sum becomes telescoping: $\sum_{r=1}^{n} t_r = \frac{1}{2}\left(\frac{1}{1} - \frac{1}{n^2 + n + 1}\right)$.</p><p>As $n \to \infty$, $\lim_{n \to \infty} \sum_{r=1}^{n} t_r = \frac{1}{2}$.</p>
Correct Answer: B