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Chapter 10: Problem 32

In Exercises \(17-46,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}} $$

Short Answer

Expert verified
The series converges by the Alternating Series Test.

Step by step solution

01

Identify the Type of Series

We are given the series \( \sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}. \) This is an alternating series since the terms switch signs due to \((-2)^n\). We can express the general term as \(a_n = \frac{n \ln n}{2^n}\) ignoring the negative sign.
02

Check the Alternating Series Test

The Alternating Series Test states that if the absolute value of the terms \(a_n\) is decreasing and \(\lim_{n \to \infty} a_n = 0\), then the series converges. We need to verify these conditions for \(a_n = \frac{n \ln n}{2^n}\).
03

Verify Decreasing Terms

To check if \( \frac{n \ln n}{2^n} \) is decreasing, consider the derivative of \( f(x) = \frac{x \ln x}{2^x} \) using the quotient rule: \(f'(x) = \frac{2^x (1 + \ln x) - x \ln x (2^x \ln 2)}{(2^x)^2}\). Simplifying this derivative shows \(f'(x) < 0\) for sufficiently large \(x\), indicating \( f(x) \) is decreasing eventually.
04

Evaluate Limit of Terms

Evaluate \( \lim_{n \to \infty} \frac{n \ln n}{2^n} \). As \( n \to \infty \), the exponential function in the denominator grows much faster than the polynomial in the numerator. Thus, \( \lim_{n \to \infty} \frac{n \ln n}{2^n} = 0 \).
05

Apply the Alternating Series Test

Since \( \frac{n \ln n}{2^n} \) is decreasing for sufficiently large \(n\) and the limit of this sequence is zero, the conditions of the Alternating Series Test are satisfied. Therefore, the series converges.

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Key Concepts

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

Series Convergence
In mathematics, when we talk about series convergence, we are essentially asking if the sum of a series has a finite value.
More intuitively, does the series "settle down" to a single number as we keep adding up its terms one by one?
  • If it does, we say the series converges.
  • If it doesn't, then it diverges and grows indefinitely or fluctuates without approaching a finite limit.
To determine whether a series converges, various tests can be used. One popular test for alternating series, like the one in the original exercise, is the Alternating Series Test. Checking these conditions helps us decide whether the infinite sum of the series approaches a fixed number as more terms are summed.
Alternating Series
An alternating series is a series whose terms alternate between positive and negative.
In an equation, we might see this alternation shown by a term like \((-1)^n\) or \((-2)^n\), where the base number induces the switch in signs.
  • Alternating series can create special patterns that are interesting to analyze because their sum might converge when non-alternating series wouldn鈥檛.
  • Each subsequent term tries to "cancel out" a little more of the series, leading to the possibility of convergence even if the individual terms don't tend to zero quickly.
In our exercise, the series \(\sum_{n=1}^{\infty} \frac{n \ln n}{(-2)^{n}}\) becomes an alternating series because of the \((-2)^n\) term.
Limiting Behavior
To decide if a series converges, particularly in the context of alternating series, we often examine the limiting behavior of its individual terms.
That is, we look at what happens to the terms as the index \(n\) gets very large.
  • The key is for these terms to get progressively smaller, at least in terms of absolute value.
  • If \(\lim_{n \to \infty} a_n = 0\), meaning the terms approach zero, it creates a scenario where adding more of these terms doesn't substantially change the total sum.
This limiting behavior is critical for the Alternating Series Test because it confirms that the series won't "explode" to an infinite sum but will stabilize.
Quotient Rule
The quotient rule is an essential calculus tool for finding the derivative of a function that is the ratio of two differentiable functions.
The quotient rule can sometimes help us check if a function is increasing or decreasing, which we need to do for testing convergence.
  • If a function that's representative of a series term is decreasing, it fits one of the criteria for certain convergence tests like the Alternating Series Test.
  • The rule is: if you have a function \(f(x) = \frac{g(x)}{h(x)}\), then its derivative is \(f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}\).
In our exercise, we used the quotient rule on \(f(x) = \frac{x \ln x}{2^x}\) to determine that \(f(x)\) is decreasing for large \(x\), which helps prove convergence via the Alternating Series Test.

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Most popular questions from this chapter

\begin{equation} \begin{array}{l}{\text { Using a CAS, perform the following steps to aid in answering }} \\ {\text { questions (a) and (b) for the functions and intervals in Exercises }} \\ {57-62 .}\\\\{\text { Step } I : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\ {x=0}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } \int^{(n+1)}(c) \text { associat- }} \\ {\text { ed with the remainder term for each Taylor polynomial. Plot }} \\ {\text { the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approxima- }} \\ {\text { tions together. Discuss the graphs in relation to the informa- }} \\ {\text { tion discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=\frac{x}{x^{2}+1}, \quad|x| \leq 2$$

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a .\) Find the (a) linearization (Taylor polynomial of order 1 ) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=\tan x\)

\begin{equation} \begin{array}{l}{\text { Using a CAS, perform the following steps to aid in answering }} \\ {\text { questions (a) and (b) for the functions and intervals in Exercises }} \\ {57-62 .}\\\\{\text { Step } I : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\ {x=0}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } \int^{(n+1)}(c) \text { associat- }} \\ {\text { ed with the remainder term for each Taylor polynomial. Plot }} \\ {\text { the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approxima- }} \\ {\text { tions together. Discuss the graphs in relation to the informa- }} \\ {\text { tion discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=(\cos x)(\sin 2 x), \quad|x| \leq 2$$

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

When \(a\) and \(b\) are real, we define \(e^{(a+i b) x}\) with the equation \begin{equation}e^{(a+i b) x}=e^{a x} \cdot e^{i b x}=e^{a x}(\cos b x+i \sin b x).\end{equation} Differentiate the right-hand side of this equation to show that \begin{equation}\frac{d}{d x} e^{(a+i b) x}=(a+i b) e^{(a+i b) x}.\end{equation} Thus the familiar rule \((d / d x) e^{k x}=k e^{k x}\) holds for \(k\) complex as well as real.
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