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We are asked to show that the series 

\[
\sum_{n=1}^{\infty} \frac{\cos(n\pi)}{n^2}
\]

is absolutely convergent.

### Step 1: Simplify the general term of the series
Notice that \(\cos(n\pi)\) takes on specific values depending on \(n\):

\[
\cos(n\pi) = (-1)^n
\]
This is because:
- \(\cos(0\pi) = 1\),
- \(\cos(\pi) = -1\),
- \(\cos(2\pi) = 1\), and so on, alternating between \(1\) and \(-1\).

Thus, the series can be rewritten as:

\[
\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}
\]

### Step 2: Test for absolute convergence
To test for absolute convergence, we need to consider the series:

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

This is the well-known *p-series* with \(p = 2\). A p-series converges if \(p > 1\), and since \(p = 2 > 1\), the series converges.

### Conclusion:
Since the series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) converges, the original series \(\sum_{n=1}^{\infty} \frac{\cos(n\pi)}{n^2}\) is absolutely convergent.

Would you like a more detailed explanation or to explore any of the steps further?

Here are 5 related questions:
1. What is a *p-series* and how does it relate to the convergence of infinite series?
2. Why is absolute convergence stronger than regular convergence for series?
3. What other convergence tests, like the ratio test, could be applied to similar series?
4. How does the alternating nature of \(\cos(n\pi)\) affect the convergence properties of the series?
5. Can we extend this method to study other series involving trigonometric terms?

**Tip:** When testing for absolute convergence, always check if the terms of the series, when made positive, correspond to a known convergent series like a p-series!

Show that the series ∑(cos(nπ)/n²) from n=1 to infinity is absolutely convergent.

Discover how to show the absolute convergence of the infinite series ∑(cos(nπ)/n²). Using the properties of trigonometric functions and the p-series test, this solution is ideal for undergraduate students studying infinite series and convergence.

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