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Chapter 13: Problem 30

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\cos n \pi$$

Short Answer

Expert verified
The sequence \(a_n = \cos(n\pi)\) is divergent because it oscillates between 1 and -1.

Step by step solution

01

Understanding the Sequence

The sequence given is \(a_n = \cos(n\pi)\). To understand this sequence, we need to examine what happens to the expression \(\cos(n\pi)\) as \(n\) takes on different integer values.
02

Evaluating the Elements of the Sequence

Calculate \(\cos(n\pi)\) for different integer values of \(n\). When \(n = 0, 1, 2, 3, 4, \ldots\), we have the results:- \(\cos(0\pi) = 1\)- \(\cos(1\pi) = -1\)- \(\cos(2\pi) = 1\)- \(\cos(3\pi) = -1\)...This creates an alternating sequence of 1, -1, 1, -1, and so on.
03

Analyzing Convergence

For a sequence to converge, the terms must approach a single value as \(n\) goes to infinity. In this case, the sequence \(1, -1, 1, -1, \ldots\) does not settle to any single value. Instead, it continues to oscillate between 1 and -1.
04

Conclusion on Convergence or Divergence

Since the sequence does not approach a single value and continues to oscillate between two values (1 and -1), it is divergent. The sequence does not converge to a limit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Alternating Sequence
An alternating sequence is a sequence of numbers that switch signs from term to term. In other words, the values of the sequence alternate between positive and negative, like a pendulum swinging back and forth.
In the original exercise, we have the sequence defined by \(a_n = \cos(n\pi)\). This sequence produces a pattern of numbers: 1, -1, 1, -1, and so forth. Since the sequence continues indefinitely without settling into a steady state, it alternates between 1 and -1 continuously.
  • Alternating sequences can sometimes converge, but only if the oscillation decreases in amplitude to zero as the sequence progresses.
  • In our example, the sequence remains constant in amplitude (1 and -1), indicating that it doesn't converge.
Understanding this behavior is crucial since it defines whether a sequence may settle into a specific value or continues swaying between fixed points, impacting its convergence or divergence.
Trigonometric Sequence
The sequence in question, \(a_n = \cos(n\pi)\), is a trigonometric sequence. Trigonometric sequences involve the sine and cosine functions, which are fundamental aspects of trigonometry.
Cosine, specifically, describes the x-coordinate on a unit circle as you rotate it by a certain angle, which implies repetition every complete turn (or \(2\pi\) radians).

  • In the context of the given sequence, the angle is \(n\pi\). This restricts the cosine values to precisely either 1 or -1 because every integer multiple of \(\pi\) lands us either at 1 or -1 on the unit circle.
  • This behavior is responsible for the consistent alternation that defines the sequence in question.
In this way, even though trigonometric sequences like this one involve periodic trigonometric functions, they reveal distinctive behaviors due to the specific angles or multipliers involved.
Limit of a Sequence
The limit of a sequence is a pivotal concept in analysis, determining whether a sequence approaches a specific number as its index heads toward infinity. Not all sequences have limits鈥攐nly those that exhibit consistency in approaching a single, well-defined value.
In the given exercise, the sequence \(a_n = \cos(n\pi)\) doesn't fulfill this requirement as its terms do not settle to one number; they oscillate between 1 and -1 endlessly.

  • The lack of convergence in this sequence implies no limit exists, thereby classifying the sequence as divergent.
  • A convergent sequence, in contrast, would have terms that inch closer together, eventually settling at a limit, like approaching zero or another constant.
This concept is essential for mathematical analysis and applications, as it influences the behavior of functions, series, or system dynamics in various fields.

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Most popular questions from this chapter

The displacement \(s\) (in meters) of a particle moving in a straight line is given by the equation of motion \(s=4 t^{3}+6 t+2,\) where \(t\) is measured in seconds. Find the instantaneous velocity of the particle \(s\) at times \(t=a\) \(t=1, t=2, t=3\)

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When we approximate areas using rectangles as in Example \(1,\) then the more rectangles we use, the more accurate the answer. The following TI- 83 program finds the approximate area under the graph of \(f\) on the interval \([a, b]\) using \(n\) rectangles. To use the program, first store the function \(f\) in \(Y_{1}\). The program prompts you to enter \(\mathrm{N}\), the number of rectangles, and \(\mathrm{A}\) and \(\mathrm{B}\), the endpoints of the interval. (a) Approximate the area under the graph of \(f(x)=x^{5}+2 x+3\) on \([1,3],\) using \(10,20\) and 100 rectangles. (b) Approximate the area under the graph of \(f\) on the given interval, using 100 rectangles. (i) \(f(x)=\sin x, \quad\) on \([0, \pi]\) (ii) \(f(x)=e^{-x^{2}}, \quad\) on \([-1,1]\) (GRAPH CANT COPY)
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