The series you're dealing with is:

$\left( \frac{1}{3} \right)^2 + \left( \frac{1}{3} \cdot \frac{2}{5} \right)^2 + \left( \frac{1}{3} \cdot \frac{2}{5} \cdot \frac{3}{7} \right)^2 + \dots$

Step 1: General term of the series

Let’s express the general term of this series. The first term is $\left( \frac{1}{3} \right)^2$, the second term is $\left( \frac{1}{3} \cdot \frac{2}{5} \right)^2$, the third term is $\left( \frac{1}{3} \cdot \frac{2}{5} \cdot \frac{3}{7} \right)^2$, and so on.

Thus, the $n$-th term (where $n$ starts from 1) of the series can be written as:

$T_n = \left( \prod_{k=1}^{n} \frac{k}{2k+1} \right)^2$

Step 2: Simplifying the general term

We can now analyze the product:

$\prod_{k=1}^{n} \frac{k}{2k+1} = \frac{1}{3} \cdot \frac{2}{5} \cdot \frac{3}{7} \cdot \dots \cdot \frac{n}{2n+1}$

Step 3: Convergence test

We need to test the convergence of the series. A useful tool for testing convergence is the ratio test, which works well with terms involving products.

Ratio test

Let’s examine the ratio of consecutive terms. Consider the ratio $\frac{T_{n+1}}{T_n}$:

$\frac{T_{n+1}}{T_n} = \frac{\left( \prod_{k=1}^{n+1} \frac{k}{2k+1} \right)^2}{\left( \prod_{k=1}^{n} \frac{k}{2k+1} \right)^2} = \left( \frac{n+1}{2(n+1)+1} \right)^2 = \left( \frac{n+1}{2n+3} \right)^2$

Now, take the limit of this ratio as $n \to \infty$:

$\lim_{n \to \infty} \frac{T_{n+1}}{T_n} = \lim_{n \to \infty} \left( \frac{n+1}{2n+3} \right)^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}$

Since the limit is less than 1 ($\frac{1}{4} < 1$), the ratio test confirms that the series converges.

Conclusion:

The series converges.

---

Would you like further details on the ratio test or any other part of the solution? Here are a few related questions to expand the topic:

1. How does the ratio test compare with other convergence tests, like the root test or the integral test?
2. What would happen if the general term was slightly different (for example, adding more terms in the product)?
3. How can we find the sum of a convergent series like this?
4. What are the properties of convergent series in terms of bounds and the behavior of partial sums?
5. How would the convergence change if the terms involved factorials or other special functions?

Tip: When working with series that involve products or factorials, breaking down the terms and using the ratio or root test can quickly help you determine convergence.