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Chapter 10: Problem 38

In Exercises \(31-38,\) use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=0}^{\infty} \cos n \pi$$

Short Answer

Expert verified
The series \( \sum_{n=0}^{\infty} \cos(n\pi) \) diverges by the nth-term test.

Step by step solution

01

Understand the Test

The nth-term test for divergence states that if \( \lim_{{n \to \infty}} a_n eq 0\), then the series \( \sum_{n=1}^{\infty} a_n \) is divergent. However, if \( \lim_{{n \to \infty}} a_n = 0\), the test is inconclusive.
02

Identify the Sequence

For the given series \( \sum_{n=0}^{\infty} \cos(n\pi) \), the sequence is \( a_n = \cos(n\pi) \). Observe that \( \cos(n\pi) \) alternates between 1 and -1 with every increment of \( n \), due to the periodic nature of the cosine function.
03

Evaluate the Limit

Evaluate the limit: \( \lim_{{n \to \infty}} \cos(n\pi) \). Since \( \cos(n\pi) = (-1)^n \), the limit does not settle to a single value; it alternates between 1 and -1 as \( n \) increases.
04

Apply the nth-Term Test

Since \( \lim_{{n \to \infty}} \cos(n\pi) \) does not equal 0, by the nth-term test for divergence, the series \( \sum_{n=0}^{\infty} \cos(n\pi) \) diverges.

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Key Concepts

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

Sequence
A sequence is a set of numbers arranged in a specific order, following a particular rule. Each number in a sequence is referred to as a term. In the context of the exercise, the sequence is derived from the expression \(a_n = \cos(n\pi)\). Here, each term of the sequence is determined by the cosine of \(n\) multiplied by \(\pi\).

In this specific sequence, there is a special alternation due to the nature of the cosine function:
  • When \(n\) is even, \(\cos(n\pi) = 1\).
  • When \(n\) is odd, \(\cos(n\pi) = -1\).
The sequence thus alternates between 1 and -1 as \(n\) increases. Understanding this alternation is crucial to analyzing the behavior of the series that it forms.
Infinite Series
An infinite series is formed by adding together all the terms of an infinite sequence. Symbolically, it is represented by the summation notation \(\sum_{n=0}^{\infty} a_n\), where \(a_n\) are the terms in a sequence.

In our exercise, the infinite series is \(\sum_{n=0}^{\infty} \cos(n\pi)\). This series consists of the sum of an infinite number of terms that alternate between 1 and -1.
  • Infinite series can either converge to a specific value or diverge.
  • The behavior depends on the properties of the sequence terms \(a_n\).
Recognizing patterns in the sequence enables us to better understand the overall tendencies of the series, which is essential for applying tests like the nth-Term Test for Divergence.
Limit
A limit is a fundamental concept in calculus that describes the value a sequence or function approaches as the input approaches some value. It is often represented as \( \lim_{{n \to \infty}} a_n \).

In the context of our exercise, we are tasked to find the limit of the sequence term \(\cos(n\pi)\) as \(n\) approaches infinity. Because \(\cos(n\pi)\) represents \((-1)^n\), it toggles between values 1 and -1:
  • If \(n\) approaches infinity, the values continue to oscillate between 1 and -1.
  • The limit \( \lim_{{n \to \infty}} \cos(n\pi) \) does not exist in the traditional sense since it does not settle at a single value.
Knowing how to evaluate these limits allows for determining whether or not the series diverges, using specific tests designed to identify such behavior.
Divergent Series
A divergent series is an infinite series that does not converge to a finite limit. This means that the sum of its terms keeps increasing indefinitely or oscillates without approaching a specific value.

In our problem, the series \(\sum_{n=0}^{\infty} \cos(n\pi)\) is a divergent series since the nth-term test for divergence indicates that:
  • The limit \( \lim_{{n \to \infty}} \cos(n\pi) eq 0 \).
  • The sequence of terms \(\cos(n\pi)\) does not diminish towards zero.
According to the nth-term test, if the limit of the sequence terms does not equal zero, the series is divergent. This means that as you sum over the terms forever, you'll not get closer to a single, finite number, but instead continue to have values that alternate or grow without bound.

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Most popular questions from this chapter

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=\frac{n^{41}}{19^{n}} $$

Prove that if \(\left\\{a_{n}\right\\}\) is a convergent sequence, then to every positive number \(\varepsilon\) there corresponds an integer \(N\) such that $$ \left|a_{m}-a_{n}\right|<\varepsilon \quad \text { whenever } \quad m>N \quad \text { and } \quad n>N $$

Find the first three nonzero terms of the Maclaurin series for each function and the values of \(x\) for which the series converges absolutely. \(f(x)=\frac{x^{3}}{1+2 x}\)

\begin{equation} \begin{array}{l}{\text { a. Use Taylor's formula with } n=2 \text { to find the quadratic }} \\ {\text { approximation of } f(x)=(1+x)^{k} \text { at } x=0(k \text { a constant })} \\ {\text { b. If } k=3, \text { for approximately what values of } x \text { in the interval }} \\ {[0,1] \text { will the error in the quadratic approximation be less }} \\ {\text { than } 1 / 100 ?}\end{array} \end{equation}

Assume that the series \(\sum a_{n} x^{n}\) converges for \(x=4\) and diverges for \(x=7 .\) Answer true \((T),\) false \((F),\) or not enough information given \((\mathrm{N})\) for the following statements about the series. $$ \begin{array}{l}{\text { a. Converges absolutely for } x=-4} \\ {\text { b. Diverges for } x=5} \\ {\text { c. Converges absolutely for } x=-8.5} \\\ {\text { d. Converges absolutely for } x=-8.5} \\ {\text { e. Diverges for } x=8} \\ {\text { f. Diverges absolutely for } x=0} \\ {\text { g. Converges absolutely for } x=-7.1}\end{array} $$
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