To solve the infinite series sum:

$\sum_{r=1}^{\infty} \frac{4r^2 - 2}{4r^4 + 1},$

we start by analyzing the general term of the series:

$a_r = \frac{4r^2 - 2}{4r^4 + 1}.$

Simplification of the Term

Let's try to factorize or simplify this expression.

1. Notice that the denominator can be rewritten as:

$4r^4 + 1 = (2r^2)^2 + 1.$

2. This suggests that the expression is not easily factorizable or directly simplifiable using standard factorization techniques.

Asymptotic Behavior for Large $r$

For large values of $r$, both $4r^2$ and $4r^4$ dominate the constants:

$a_r \approx \frac{4r^2}{4r^4} = \frac{1}{r^2}.$

This suggests that the terms of the series behave similarly to $\frac{1}{r^2}$ for large $r$, hinting that the series might converge (since $\sum \frac{1}{r^2}$ is a convergent p-series with $p = 2 > 1$).

Convergence Test

Let's formally test the convergence of the series. One way to do this is by using the Comparison Test or the Limit Comparison Test with $\sum \frac{1}{r^2}$.

Limit Comparison Test with $\sum \frac{1}{r^2}$:

$\lim_{r \to \infty} \frac{a_r}{\frac{1}{r^2}} = \lim_{r \to \infty} \frac{\frac{4r^2 - 2}{4r^4 + 1}}{\frac{1}{r^2}} = \lim_{r \to \infty} \frac{4r^4 - 2r^2}{4r^4 + 1}.$

Simplifying further:

$= \lim_{r \to \infty} \frac{4 - \frac{2}{r^2}}{4 + \frac{1}{r^4}} = \frac{4 - 0}{4 + 0} = 1.$

Since the limit is a positive finite number, the Limit Comparison Test confirms that both series $\sum a_r$ and $\sum \frac{1}{r^2}$ either both converge or both diverge. As $\sum \frac{1}{r^2}$ is a convergent series, $\sum_{r=1}^\infty \frac{4r^2 - 2}{4r^4 + 1}$ also converges.

Conclusion

The series:

$\sum_{r=1}^{\infty} \frac{4r^2 - 2}{4r^4 + 1}$

converges. However, finding the exact sum might be quite complicated and would typically require more advanced techniques or numerical approximation methods.

Would you like more details on any step or a further discussion of the series' sum?

Further Related Questions:

1. What is the exact value of the series sum, if computable?
2. How can we approximate the sum numerically?
3. What is the convergence behavior of the series when compared to other p-series?
4. Can we find the partial sum formula for this series?
5. What are the general methods to determine the convergence of series like this?

Tip:

Always consider comparing a given series to a known convergent or divergent series to use comparison tests effectively.