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Chapter 9: Problem 35

Determining Convergence or Divergence In Exercises \(33-36,\) use the polynomial test given in Exercise 32 to determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{1}{n^{3}+1} $$

Short Answer

Expert verified
The given series \(\sum_{n=1}^{\infty} \frac{1}{n^{3}+1}\) converges.

Step by step solution

01

Understanding the Polynomial Test for Convergence/Divergence

The polynomial test for convergence or divergence is a simple concept based on the ratio of the polynomial degrees. The resulting limit of the ratio of two sequences as n approaches infinity provides information on the behavior of the series. If the degree of the denominator is greater than the degree of the numerator, the series converges. Otherwise, it diverges.
02

Apply the Polynomial Test

First, observe the series and identify the numerator and denominator. Here, our series is \(\sum_{n=1}^{\infty} \frac{1}{n^{3}+1}\) which is equivalent to \(\sum_{n=1}^{\infty} \frac{n^{0}}{n^{3}+1}\). The degree of the polynomial in the numerator is 0, and the degree of the polynomial in the denominator is 3. So, the degree of the denominator is greater than the degree of the numerator.
03

Determine the Convergence or Divergence

As per the polynomial test for convergence, if the degree of the denominator is greater than the degree of the numerator, the series converges. As such, the given series \(\sum_{n=1}^{\infty} \frac{1}{n^{3}+1}\) converges.

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Key Concepts

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

Convergence of Series
Understanding the concept of convergence is crucial when working with series. When we speak of the convergence of a series, we refer to the idea that as we add more and more terms, the sum approaches a certain finite value. In simpler terms, the series 'settles down' to a specific number as the number of terms grows larger.

This property is not true for all series. For instance, the sum of the series 1 + 2 + 3 + 4 + ... grows without bound as more terms are added; hence, this series does not converge, it diverges. However, a series like \(\frac{1}{2}\) + \(\frac{1}{4}\) + \(\frac{1}{8}\) + ... does approach a finite value, which is 1 in this case. Therefore, it is said to converge.

To determine convergence, several tests can be applied. The polynomial test, as seen in the exercise, is one such method and is based on comparing the degrees of the numerator and denominator of a term within the series.
Infinite Series
An infinite series is a sum of infinitely many terms. Each term in the series is typically generated based on a certain rule or formula. A simple example is the geometric series \(1 + r + r^2 + r^3 + ...\), where each term is a fixed multiple 'r' of the previous term.

The series in our exercise, \(\frac{1}{n^3+1}\), is an infinite series because it continues indefinitely, with 'n' taking on every positive integer value. Despite this infinity of terms, it can still converge. To explore this paradoxical sounding situation, mathematicians use limits to assess what value, if any, the series settles towards as 'n' approaches infinity. This leads us to various tests for convergence, like the polynomial test, to determine whether we get a finite limit.
Degree of Polynomial
The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial \(5x^3 - x^2 + 6\), the highest power of 'x' is 3, so the degree of the polynomial is 3.

Knowing the degree comes into play prominently when assessing the behavior of the terms in a series, such as in the polynomial test for convergence. In our exercise \(\frac{1}{n^3+1}\), the highest power of 'n' in the denominator is 3, so the polynomial's degree is 3. This information is key to determining the convergence of the series because, as stated in the solution, if the degree of the denominator is higher than the numerator, it suggests that as 'n' grows larger, the terms of the series decrease to zero, allowing the series to potentially converge.

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Most popular questions from this chapter

(a) determine the number of terms required to approximate the sum of the series with an error less than 0.0001, and (b) use a graphing utility to approximate the sum of the series with an error less than 0.0001. $$\sum_{k=0}^{\infty} \frac{(-3)^{k}}{1 \cdot 3 \cdot 5 \cdot \cdot(2 k+1)}$$

Determine the convergence or divergence of the series $$\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(x n) !}$$ when (a) \(x=1,\) (b) \(x=2,(\mathrm{c}) x=3,\) and \((\mathrm{d}) x\) is a positive integer.

Evaluating a Binomial Coefficient In Exercises \(89-92\) , evaluate the binomial coefficient using the formula $$\left(\begin{array}{l}{k} \\ {n}\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdot \cdot(k-n+1)}{n !}$$ where \(k\) is a real number, \(n\) is a positive integer, and \(\left(\begin{array}{l}{k} \\ {0}\end{array}\right)=1\) $$ \left(\begin{array}{c}{-1 / 3} \\ {5}\end{array}\right) $$

Investigation Consider the function \(f\) defined by $$f(x)=\left\\{\begin{array}{ll}{e^{-1 / x^{2}},} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.$$ (a) Sketch a graph of the function. (b) Use the alternative form of the definition of the derivative (Section 2.1) and L'H贸pital's Rule to show that \(f^{\prime}(0)=0\) . [By continuing this process, it can be shown that \(f^{(n)}(0)=0\) for \(n>1 . ]\) (c) Using the result in part (b), find the Maclaurin series for \(f\) . Does the series converge to \(f ?\)

Projectile Motion A projectile fired from the ground follows the trajectory given by $$y=\left(\tan \theta-\frac{g}{k v_{0} \cos \theta}\right) x-\frac{g}{k^{2}} \ln \left(1-\frac{k x}{v_{0} \cos \theta}\right)$$ where \(v_{0}\) is the initial speed, \(\theta\) is the angle of projection, \(g\) is the acceleration due to gravity, and \(k\) is the drag factor caused by air resistance. Using the power series representation $$\ln (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots, \quad-1 < x < 1$$ verify that the trajectory can be rewritten as $$y=(\tan \theta) x+\frac{g x^{2}}{2 v_{0}^{2} \cos ^{2} \theta}+\frac{k g x^{3}}{3 v_{0}^{3} \cos ^{3} \theta}+\frac{k^{2} g x^{4}}{4 v_{0}^{4} \cos ^{4} \theta}+\cdots$$
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