To prove the convergence of the series

$\sum_{n=1}^{\infty} \frac{(-1)^{n-1} \cos^2(nx)}{n \sqrt{n}},$

we need to investigate its convergence properties using appropriate convergence tests, such as the Alternating Series Test, the Absolute Convergence Test, or other suitable methods.

Step 1: Analyzing the General Term

The general term of the series is given by:

$a_n = \frac{(-1)^{n-1} \cos^2(nx)}{n \sqrt{n}}.$

The sequence alternates in sign due to the $(-1)^{n-1}$ term, and the magnitude of the term is $\frac{\cos^2(nx)}{n \sqrt{n}}$.

Step 2: Testing Absolute Convergence

To test absolute convergence, we examine the series without the alternating sign:

$\sum_{n=1}^{\infty} \left| a_n \right| = \sum_{n=1}^{\infty} \frac{\cos^2(nx)}{n \sqrt{n}}.$

Since $\cos^2(nx)$ oscillates between 0 and 1, we have:

$0 \leq \cos^2(nx) \leq 1 \implies 0 \leq \frac{\cos^2(nx)}{n \sqrt{n}} \leq \frac{1}{n \sqrt{n}}.$

Step 3: Applying the Comparison Test

We compare the series $\sum \frac{\cos^2(nx)}{n \sqrt{n}}$ with the series $\sum \frac{1}{n \sqrt{n}}$.

1. Consider the series:

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

2. This is a p-series with $p = \frac{3}{2} > 1$, which converges.

Since $\frac{\cos^2(nx)}{n \sqrt{n}} \leq \frac{1}{n \sqrt{n}}$ and $\sum \frac{1}{n^{3/2}}$ converges, by the Comparison Test, the series $\sum \frac{\cos^2(nx)}{n \sqrt{n}}$ also converges.

Step 4: Applying the Alternating Series Test

To confirm the convergence of the original series, we use the Alternating Series Test, which requires:

1. Decreasing Magnitude: Check if $\frac{\cos^2(nx)}{n \sqrt{n}}$ is decreasing.

   Since $\cos^2(nx)$ is bounded and the term $\frac{1}{n \sqrt{n}}$ decreases as $n$ increases, the product decreases overall.

2. Limit to Zero: Check if $\lim_{n \to \infty} \frac{\cos^2(nx)}{n \sqrt{n}} = 0$.

   This limit is zero since $\frac{1}{n \sqrt{n}} \to 0$ as $n \to \infty$.

Since both conditions are met, the series converges by the Alternating Series Test.

Conclusion

The series

$\sum_{n=1}^{\infty} \frac{(-1)^{n-1} \cos^2(nx)}{n \sqrt{n}}$

converges.

Would you like further details or have any questions?

Related Questions

1. How does the Alternating Series Test work in general?
2. What is the p-series test, and how is it used?
3. Can a series converge conditionally but not absolutely?
4. What are some examples of other convergence tests?
5. How does the behavior of $\cos(nx)$ affect the convergence of series?

Tip: When proving series convergence, always consider testing both absolute and conditional convergence for a complete analysis.