Question

For the following sequences, plot the first 25 terms of the sequence and state whether the graphical evidence suggests that the sequence converges or diverges. $$ \text { [T] } a_{n}=\cos n $$

Short answer

The sequence diverges; it does not converge to a particular value.

Step-by-step solution

Understand the Sequence Function

The sequence is given as \( a_n = \cos n \), where \( n \) represents the term number in the sequence. Here, \( n \) takes integer values starting from 1. This sequence involves the cosine function, which is a periodic trigonometric function.

Generate the Sequence Terms

Compute the first 25 terms of the sequence by substituting values \( n = 1, 2, 3, \ldots, 25 \) into the function \( a_n = \cos n \). Here, \( \cos n \) will output values between -1 and 1, and these values will oscillate as \( n \) increases.

Plot the Sequence

Create a plot for the first 25 terms of the sequence \( a_n = \cos n \). On the x-axis, use \( n \, (1 \leq n \leq 25) \), and on the y-axis, plot the corresponding \( a_n \) values. You will observe oscillating points throughout the range of \( n \).

Analyze the Plot for Convergence or Divergence

Examine the plotted sequence. The nature of \( \cos n \) is such that it does not approach a single fixed value as \( n \) increases; instead, it continues to oscillate. The points scatter between -1 and 1 without settling on a particular value.

Key concepts

Trigonometric Sequences

When exploring mathematical sequences, trigonometric sequences can often appear mysterious yet fascinating. These sequences comprise terms that involve trigonometric functions, such as sine or cosine. In this exercise, we encounter the sequence \( a_n = \cos n \). Trigonometric sequences have unique properties because trigonometric functions are periodic, typically with a period of \( 2\pi \), meaning they repeat their values in a predictable pattern over time.
This periodic nature significantly influences the sequence's behavior, especially concerning aspects such as convergence or divergence. Convergence refers to the situation where the terms of the sequence approach a specific value as the term number \( n \) becomes very large. In contrast, divergence implies the sequence lacks such a tendency. Understanding these terms helps set the stage for analyzing the behavior of specific types like cosine sequences.

Cosine Function

The cosine function, denoted as \( \cos x \), is a fundamental trigonometric function crucial in both sequences and periodic analysis. Defined as the x-coordinate of the point on the unit circle at an angle \( x \) from the positive x-axis, the cosine of an angle describes a wave-like pattern.
Its periodicity makes it an essential function for oscillating phenomena, represented by a repeating pattern along the y-axis between -1 and 1.

• The standard period for \( \cos x \) is \( 2\pi \).
• It has an amplitude of 1, indicating its maximum deviation from the central axis.
• The function is even, so \( \cos(-x) = \cos(x) \).

Understanding these properties of the cosine function helps us predict and analyze the sequence \( a_n = \cos n \). Because the integers \( n \) skip entire periods, they lead to seemingly erratic, non-repetitive points when plotted, impacting our convergence or divergence assessment.

Graphical Analysis of Sequences

Graphical analysis offers a valuable tool for comprehending sequence behavior, especially trigonometric sequences like \( a_n = \cos n \). By plotting the sequence's first 25 terms, we can visually interpret their behavior.
When analyzing the graph of \( a_n = \cos n \):

• The x-axis represents the term number \( n \),
• The y-axis indicates the cosine value at each \( n \).
• The points appear scattered due to the non-increasing sequentially odd intervals of \( n \).
• Each value ranges between -1 and 1 reflecting the nature of the cosine function.

Such visual analysis highlights the sequence’s oscillating nature without settling into a fixed pattern or value as \( n \) increases. This consistent oscillation leads us to conclude that the sequence does not converge but diverges. While the graph doesn’t show a narrowing path towards a single value, it effectively communicates non-convergence through its irregular scatter.