91Ó°ÊÓ

The explicit formula is used to find the terms of a sequence without needing any preceding terms. It directly expresses each term as a function of its position number (n). In our exercise, the explicit formula is \(a_{n} = \frac{n \cos(n\pi)}{2n - 1}\). This formula tells us how each term relates to \(n\) directly, which is helpful because:

• It streamlines calculations for terms at any position \(n\).
• Only the formula is needed to compute the terms of the sequence.

For instance, by simply substituting \(n = 1, 2, 3, 4, 5\) into the formula, you obtain the terms \(-1, \frac{2}{3}, -\frac{3}{5}, \frac{4}{7}, -\frac{5}{9}\). Each calculation is straightforward and the formula enables us to predict the behavior of the sequence further out.

An alternating sequence is a sequence of numbers whose terms alternate in sign. Here, the sequence \(a_{n} = \frac{n \cos(n\pi)}{2n - 1}\) is an example.

• For even \(n\), \(\cos(n\pi) = 1\), making the term positive.
• For odd \(n\), \(\cos(n\pi) = -1\), making the term negative.

This sign change pattern creates an alternating sequence. The fluctuations between negative and positive values impact whether a sequence converges. It's crucial to understand alternating sequences when analyzing convergence because the alternating nature can heavily influence the sequence's behavior over time.

Limit analysis helps determine whether a sequence converges towards a single finite value as \(n\) approaches infinity. To examine the sequence \(a_{n} = \frac{n \cos(n\pi)}{2n - 1}\), we consider\[\lim_{{n \to \infty}} \frac{n \cos(n\pi)}{2n - 1}\]Firstly, focusing on the magnitude, ignoring sign,\[\left| \frac{n}{2n-1} \right| = \left| \frac{1}{2 - \frac{1}{n}} \right|\]as \(n\) approaches infinity, this expression approaches \(\frac{1}{2}\). Although the magnitude suggests convergence towards \(\frac{1}{2}\), the varied sign due to \(\cos(n\pi)\) causes the sequence values to alternate. Thus, the alternating signs hinder the sequence from settling on a single value, demonstrating that \(a_{n}\) diverges. Proper limit analysis considers both magnitude and sign variation to conclude convergence or divergence.