The given series is:

$\sum_{n=1}^\infty \left| \frac{(-1)^{n-1}}{n^2} \right|$

We will analyze its convergence:

Step 1: Examine the general term

The general term of the series is:

$\left| \frac{(-1)^{n-1}}{n^2} \right| = \frac{1}{n^2}.$

This is a series of non-negative terms because we are taking the absolute value of each term.

Step 2: Check for convergence

The series becomes:

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

This is a well-known p-series, where $p = 2 > 1$. A p-series with $p > 1$ converges. Therefore, the series converges.

Step 3: Final Answer

The series converges absolutely.

Would you like further details about the convergence of p-series or related examples?

Here are some related questions for deeper exploration:

1. What is the difference between absolute and conditional convergence of a series?
2. How does the comparison test apply to p-series?
3. Could the series $\sum_{n=1}^\infty \frac{1}{n}$ converge, and why?
4. Can we compute the exact value of $\sum_{n=1}^\infty \frac{1}{n^2}$?
5. How does the alternating series test relate to this problem?

Tip: For p-series $\sum \frac{1}{n^p}$, convergence depends solely on whether $p > 1$.