prove the series Σ(-1)^n-1.cos^2 nx/ns(sqrt(n))

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.

prove the series Σ(-1)^n-1.cos^2(nx)/n(sqrt(n))

Learn how to prove the convergence of the alternating series Σ(-1)^n-1 * cos^2(nx) / (n * sqrt(n)) using the Alternating Series Test, Comparison Test, and p-series Test. Detailed steps and mathematical insights are provided, suitable for undergraduate calculus students.

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