The expression you have provided is the infinite series:

$\sum_{n=1}^{\infty} \frac{\sqrt{2n+1}}{n^2}$

To analyze and determine if this series converges or diverges, let's explore convergence tests that could apply here.

Step 1: Asymptotic Behavior

We first need to study the behavior of the general term:

$a_n = \frac{\sqrt{2n+1}}{n^2}$

For large $n$, the term $2n+1$ behaves similarly to $2n$, so the term can be approximated as:

$a_n \sim \frac{\sqrt{2n}}{n^2} = \frac{\sqrt{2} \cdot \sqrt{n}}{n^2} = \frac{\sqrt{2}}{n^{3/2}}$

Step 2: Apply a Comparison Test

Since for large $n$, $a_n \sim \frac{C}{n^{3/2}}$ (where $C$ is a constant), we compare the series with the known convergent p-series:

$\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$

The p-series converges if $p > 1$. In this case, since $\frac{3}{2} > 1$, the comparison test suggests that the given series converges.

Conclusion:

The series $\sum_{n=1}^{\infty} \frac{\sqrt{2n+1}}{n^2}$ converges based on the asymptotic comparison with the $\frac{1}{n^{3/2}}$ p-series.

Would you like more detailed steps, or do you have any questions?

Related Questions:

1. How can we use the ratio test to verify the convergence of this series?
2. What are p-series, and how are they used in convergence tests?
3. Can this series be integrated using any numerical methods?
4. How does the comparison test work for different types of series?
5. What is the difference between absolute and conditional convergence?

Tip:

For any infinite series, analyzing its behavior at infinity is key to determining convergence or divergence through comparison tests, ratio tests, or integral tests.