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Chapter 10: Problem 37

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{3^{n-1}+1}\end{equation}

Short Answer

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

Step by step solution

01

Identify a Suitable Convergence Test

To determine the convergence or divergence of the series \( \sum_{n=1}^{\infty} \frac{1}{3^{n-1}+1} \), we will use the Limit Comparison Test. This test is suitable here because \( \frac{1}{3^{n-1}+1} \) resembles a geometric series which is a well-known comparison series.
02

Choose a Comparison Series

We select the geometric series \( \sum_{n=1}^{\infty} \frac{1}{3^n} \) to compare with our series. This geometric series has a common ratio \( r = \frac{1}{3} \), where \( |r| < 1 \), so it converges.
03

Apply the Limit Comparison Test

To apply the Limit Comparison Test, compute the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \) where \( a_n = \frac{1}{3^{n-1}+1} \) and \( b_n = \frac{1}{3^n} \). Simplify and compute:\[L = \lim_{n \to \infty} \frac{\frac{1}{3^{n-1}+1}}{\frac{1}{3^n}} = \lim_{n \to \infty} \frac{3^n}{3^{n-1} + 1} = \lim_{n \to \infty} \frac{3}{1 + \frac{1}{3^{n-1}}} = 3\]The limit \( L = 3 \) which is positive and finite.
04

Conclude with the Limit Comparison Test

Since the limit \( L = 3 \) is a positive finite number, and the comparison series \( \sum_{n=1}^{\infty} \frac{1}{3^n} \) converges, the Limit Comparison Test tells us that the series \( \sum_{n=1}^{\infty} \frac{1}{3^{n-1}+1} \) also converges.

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Key Concepts

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

limit comparison test
The Limit Comparison Test is a powerful tool for determining the convergence or divergence of an infinite series. It's often used when a series resembles another, more familiar series whose convergence properties are known. The core idea is to compare the original series to a simpler one by taking the limit of their ratio.
Let鈥檚 break it down:
  • Choose two series, for instance, \( \sum a_n \) and \( \sum b_n \), where \( b_n \) is a series with known convergence.
  • Compute the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \).
If the limit \( L \) is a positive finite number, then both series will either converge or diverge together. It is crucial to select a comparison series with similar behavior to ensure the test reflects true convergent properties.
geometric series
A geometric series is a series of the form \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio. The series solely depends on the value of \( |r| \) for its convergence:
  • If \( |r| < 1 \), the series converges to the sum \( \frac{a}{1-r} \).
  • If \( |r| \geq 1 \), the series diverges.
A notable advantage of geometric series is their simplicity and the straightforward conditions which dictate their convergence. This makes them a popular choice for comparison in convergence tests, such as the Limit Comparison Test, especially when the series in question takes a simple exponential form.
series convergence
Series convergence is crucial in the study of infinite series. It refers to whether the sum of an infinite series settles down to a finite number or not. There are several tests available to determine convergence, with each test suitable for different scenarios:
  • Limit Comparison Test: Ideal when comparing to a geometric or p-series.
  • Ratio Test: Useful for series involving factorials or exponentials.
  • Integral Test: Particularly handy when the series terms are positive and resemble a function that can be integrated.
Understanding when and how to apply these tests can help you identify the convergence of a series effectively. Each test has specific criteria and conditions under which they yield correct conclusions about convergence.
infinite series
An infinite series is the sum of an infinite sequence of terms. Formally, it can be expressed as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) are the terms of the sequence. The key interest in examining such series is to determine whether they converge to a finite value or diverge to infinity or somewhere undefined.
Considerations include:
  • Convergence Tests: Techniques like the Limit Comparison, Ratio, and Integral tests help determine the behavior of these sums.
  • Types of Series: Includes geometric series, p-series, harmonic series, etc.
  • Application: Convergent series find applications in calculus, physics, and finance for approximating complex phenomena.
Understanding infinite series requires practice, patience, and learning to choose appropriate tests and series for comparison.

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Most popular questions from this chapter

Computer Explorations Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: \begin{equation} \begin{array}{l}{\text { a. For what values of } x \text { can the function be replaced by each }} \\ {\text { approximation with an error less than } 10^{-2} \text { ? }} \\ {\text { b. What is the maximum error we could expect if we replace the }} \\ {\text { function by each approximation over the specified interval? }}\end{array} \end{equation} Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises \(53-58 .\) \begin{equation} \begin{array}{l}{\text { Step } 1 : \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 } f^{(n+1)}(c) \text { associated }} \\ {\text { with the remainder term for each Taylor polynomial. }} \\ {\text { Plot 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 approximations }} \\ {\text { together. Discuss the graphs in relation to the information }} \\ {\text { discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=(1+x)^{3 / 2},-\frac{1}{2} \leq x \leq 2$$

Use series to evaluate the limits. \begin{equation} \lim _{y \rightarrow 0} \frac{y-\tan ^{-1} y}{y^{3}} \end{equation}

Use series to estimate the integrals' values with an error of magnitude less than \(10^{-5}\) . (The answer section gives the integrals' values rounded to seven decimal places.) \begin{equation} \int_{0}^{0.35} \sqrt[3]{1+x^{2}} d x \end{equation}

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{(n-1) !}{(n+2) !}\end{equation}

Euler's constant Graphs like those in Figure 10.11 suggest that as \(n\) increases there is little change in the difference between the sum $$ 1+\frac{1}{2}+\dots+\frac{1}{n} $$ and the integral $$ \ln n=\int_{1}^{n} \frac{1}{x} d x $$ To explore this idea, carry out the following steps. a. By taking \(f(x)=1 / x\) in the proof of Theorem \(9,\) show that $$ \ln (n+1) \leq 1+\frac{1}{2}+\cdots+\frac{1}{n} \leq 1+\ln n $$ or $$ 0<\ln (n+1)-\ln n \leq 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n \leq 1 $$ Thus, the sequence $$ a_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n $$ is bounded from below and from above. b. Show that $$ \frac{1}{n+1}<\int_{n}^{n+1} \frac{1}{x} d x=\ln (n+1)-\ln n $$ and use this result to show that the sequence \(\left\\{a_{n}\right\\}\) in part (a) is decreasing. since a decreasing sequence that is bounded from below converges, the numbers \(a_{n}\) defined in part (a) converge: $$ 1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n \rightarrow \gamma $$ The number \(\gamma,\) whose value is 0.5772\(\ldots\) is called Euler's constant.
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