The problem involves finding the limit of a series as $n \to \infty$. Let's analyze and simplify the series.

Problem Breakdown:

The given series is:

$\lim_{n \to \infty} \left( \frac{3}{5} + \frac{3}{25} - \cdots + 3 \cdot \frac{(-1)^n}{5^n} \right).$

This is a geometric series with the general term:

$a_n = 3 \cdot \frac{(-1)^n}{5^n}.$

The first term is $\frac{3}{5}$, and the common ratio is $r = -\frac{1}{5}$.

Infinite Geometric Series Formula:

For an infinite geometric series with first term $a$ and common ratio $r$, where $|r| < 1$, the sum is given by:

$S = \frac{a}{1 - r}.$

Applying to the Series:

1. First term: $a = \frac{3}{5}$,
2. Common ratio: $r = -\frac{1}{5}$.

Substituting into the formula:

$S = \frac{\frac{3}{5}}{1 - \left(-\frac{1}{5}\right)} = \frac{\frac{3}{5}}{1 + \frac{1}{5}} = \frac{\frac{3}{5}}{\frac{6}{5}} = \frac{3}{6} = \frac{1}{2}.$

Final Answer:

$\lim_{n \to \infty} \left( \frac{3}{5} + \frac{3}{25} - \cdots + 3 \cdot \frac{(-1)^n}{5^n} \right) = \frac{1}{2}.$

Would you like further details on the steps or explanation?

Here are 5 related questions for deeper exploration:

1. How does the sum of a finite geometric series differ from an infinite geometric series?
2. What conditions must hold for an infinite geometric series to converge?
3. Can you find the sum of the series if the common ratio $r$ is greater than or equal to 1?
4. How do alternating signs in the terms (due to $(-1)^n$) affect the convergence of a series?
5. How would this series change if the first term were different?

Tip: When working with infinite series, always check the magnitude of the common ratio $|r|$ to determine convergence.