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Chapter 10: Problem 34

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

Short Answer

Expert verified
The series \(\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^{2}+1}\) converges.

Step by step solution

01

Compare with Standard Series

To determine convergence or divergence, we need to compare the given series with a known p-series. The given series is \( \sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^{2}+1} \). Notice that \( \sqrt{n} = n^{1/2} \). For large n, \( n^2+1 \approx n^2 \), so we approximate \( \frac{\sqrt{n}}{n^2+1} \approx \frac{n^{1/2}}{n^2} = \frac{1}{n^{3/2}} \). This approximation resembles the p-series \( \sum \frac{1}{n^{p}} \) with \( p = \frac{3}{2} \).
02

Determine Behavior of the Comparison Series

Recall that a p-series \( \sum \frac{1}{n^p} \) converges if \( p > 1 \), and diverges if \( p \leq 1 \). Here, \( p = \frac{3}{2} > 1 \), implying that the comparison p-series \( \sum \frac{1}{n^{3/2}} \) converges.
03

Apply the Limit Comparison Test

To apply the Limit Comparison Test, consider the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \), where \( a_n = \frac{\sqrt{n}}{n^2+1} \) and \( b_n = \frac{1}{n^{3/2}} \). Hence, \[ L = \lim_{n \to \infty} \frac{\frac{\sqrt{n}}{n^2+1}}{\frac{1}{n^{3/2}}} = \lim_{n \to \infty} \frac{n^{3/2} \cdot \sqrt{n}}{n^2+1} = \lim_{n \to \infty} \frac{n^2}{n^2+1}. \]
04

Evaluate the Limit

The limit \( L = \lim_{n \to \infty} \frac{n^2}{n^2+1} = \lim_{n \to \infty} \frac{1}{1+\frac{1}{n^2}} \) converges to 1 as \( n \to \infty \). Since \( 0 < L < \infty \), and since the comparison series \( \sum \frac{1}{n^{3/2}} \) converges, the Limit Comparison Test confirms that \( \sum \frac{\sqrt{n}}{n^2+1} \) also converges.

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Key Concepts

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

p-series
A p-series is one of the simplest types of series we encounter in mathematics, particularly when studying convergence. A p-series is expressed as \( \sum \frac{1}{n^p} \), where \( p \) is a constant and \( n \) is a positive integer. Understanding the behavior of a p-series is key to analyzing many other series.
  • If \( p > 1 \), the series converges. This means the sum approaches a finite number as we include more terms.
  • If \( p \leq 1 \), the series diverges. In this case, the sum continues to grow without bound.
For example, in the solution, the series \( \sum \frac{1}{n^{3/2}} \) is identified as a p-series with \( p = \frac{3}{2} \). Since \( \frac{3}{2} > 1 \), it confirms that this p-series converges.
limit comparison test
The Limit Comparison Test is a powerful tool used to determine the convergence or divergence of a series by comparing it to another series whose behavior is known. Here's how it works:
  • We have two series: \( \sum a_n \) and \( \sum b_n \), with terms \( a_n \) and \( b_n \) respectively.
  • We calculate the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \).

Application and Evaluation

  • If the limit \( 0 < L < \infty \), both series either converge or diverge together.
  • If \( L = 0 \) and \( \sum b_n \) converges, then \( \sum a_n \) also converges.
  • If \( L = \infty \) and \( \sum b_n \) diverges, then \( \sum a_n \) also diverges.
In the original exercise, the series \( \sum \frac{\sqrt{n}}{n^2+1} \) is analyzed using the Limit Comparison Test with \( \sum \frac{1}{n^{3/2}} \). By calculating \( L = 1 \), and knowing that the comparison series converges, we establish the convergence of the original series.
convergence and divergence of series
Determining when a series converges or diverges is a fundamental part of calculus and analysis. A convergent series is one where the sum of its terms approaches a specific finite value as more terms are added. Conversely, a divergent series is one where the sum does not approach any finite limit.

Key Indicators

  • A quick way to judge convergence is by comparing a series to another well-known series, such as a p-series.
  • Tests like the Limit Comparison Test provide a systematic method to evaluate series, especially when direct comparison is challenging.
  • Understanding the underlying growth of terms, such as \( n^{1/2} \) approximately simplifying to a p-series, helps in identifying applicable tests.
In the problem, convergence of \( \sum \frac{\sqrt{n}}{n^2+1} \) was shown by comparing it to a convergent p-series. Each series offers unique insights and with the right tools, we can robustly determine their nature.

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Most popular questions from this chapter

Improving approximations of \(\pi\) \begin{equation} \begin{array}{l}{\text { a. Let } P \text { be an approximation of } \pi \text { accurate to } n \text { decimals. Show }} \\ {\text { that } P+\sin P \text { gives an approximation correct to } 3 n \text { decimals. }} \\ {\text { (Hint: Let } P=\pi+x . )} \\ {\text { b. Try it with a calculator. }}\end{array} \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{\operatorname{coth} n}{n^{2}}\end{equation}

Use series to evaluate the limits. \begin{equation} \lim _{x \rightarrow \infty}(x+1) \sin \frac{1}{x+1} \end{equation}

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} $$f(x)=e^{-x} \cos 2 x, \quad|x| \leq 1$$
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