Introduction

Understanding the convergence of series is a fundamental aspect of mathematical analysis. This article delves into the analysis of the convergence of the series (sum_{n1}^{infty} frac{cos{n}}{n^p}). We will explore both conditional and absolute convergence, providing a detailed explanation of the methods used and the results obtained.

Conditions for Convergence

First, we need to consider the initial condition for the exponent (p). For (p 0), the (n)th term clearly does not converge to zero, as the term itself does not exist. Therefore, we only consider (p 0).

Applying Dirichlet’s Test for Convergence

Step 1: Defining Sequences

We define (a_n frac{1}{n^p}) and (b_n cos{n}). The sequence (a_n) is a decreasing sequence that converges to zero for (p 0).

Step 2: Boundedness of Partial Sums of (b_n)

To prove that (b_n) has bounded partial sums, we use the finite geometric series formula:

(sum_{n1}^N e^{ni} frac{e^i 1 - e^{Ni}}{1 - e^i})

Taking the real part, we get:

(text{Re} left(frac{e^i 1 - e^{Ni}}{1 - e^i}right))

We can bound this as follows:

(left|text{Re}left(frac{e^i 1 - e^{Ni}}{1 - e^i}right)right| leq frac{2}{1 - e^i})

Since (frac{2}{1 - e^i}) is a constant independent of (N), the partial sums of (b_n) are indeed bounded.

Conclusion Using Dirichlet’s Test

Thus, the given series converges for any (p 0) by Dirichlet’s Test for convergence.

Absolute Convergence

(sum_{n1}^{infty} frac{cos{n}}{n^p}) is absolutely convergent for any (p 1) because (cos{n}) has amplitude one, and applying the Direct Comparison Test:

(left|frac{cos{n}}{n^p}right| leq frac{1}{n^p})

and the series (sum_{n1}^{infty} frac{1}{n^p}) converges for any (p 1).

Conditional Convergence

(sum_{n1}^{infty} frac{cos{n}}{n^p}) is conditionally convergent for (0 p 1). To see this, note that:

(left|frac{cos{n}}{n^p}right| geq frac{cos^2{n}}{n^p} frac{1 - cos{2n}}{2n^p})

Since (sum_{n1}^{infty} frac{cos{2n}}{2n^p}) is convergent for (0 p 1), but (sum_{n1}^{infty} frac{1}{2n^p}) is divergent (being a nonzero constant multiple of a divergent p-series for (p 1)), the series (sum_{n1}^{infty} frac{1 - cos{2n}}{2n^p}) diverges. Hence, by the Direct Comparison Test, the series (sum_{n1}^{infty} frac{cos{n}}{n^p}) diverges for (0 p 1) and thus is conditionally convergent for (0 p 1).

Conclusion

The analysis of the series (sum_{n1}^{infty} frac{cos{n}}{n^p}) reveals that it converges for any (p 0), and is absolutely convergent for (p 1). For (0 p 1), it is conditionally convergent. The methods used include Dirichlet’s Test and the Direct Comparison Test, providing a comprehensive understanding of its behavior.