/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Determine whether the statement ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 55

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. Suppose that \(f\) is a continuous, positive, and decreasing function on $[1, \infty) .\( If \)f(n)=a_{n}\( for \)n \geq 1\( and \)\sum_{n=1}^{\infty} a_{n}$ is convergent, then $\sum_{n=1}^{\infty} a_{n} \leq a_{1}+\int_{1}^{\infty} f(x) d x$.

Short Answer

Expert verified
The statement is true. We established that for each \(n\), the height of the rectangle is given by \(f(x)\geq a_{n+1}\), and thus the sum of the series \(\sum_{n=1}^{\infty} a_n\) is less than or equal to the sum of the area under the curve of the function \(f(x)\), represented as \(a_1 + \int_{1}^{\infty} f(x) dx\).

Step by step solution

01

First, let's think about the area under the curve \(f(x)\) that can be covered by rectangles of length 1. Each rectangle can be represented as \(a_n\) with \(n \geq 1\). Since \(f\) is positive and decreasing, the rectangles get smaller as we continue along the x-axis. Notice that for any value \(x \in [n,n+1)\) the height of the rectangle is given by \(f(x) \geq f(n+1) = a_{n+1}\). #Step 2: Compare the series and areas of the rectangles#

Let's compare the sum of the series and the areas of the rectangles: $$\sum_{n=1}^{\infty} a_n \leq a_1 + \sum_{n=1}^{\infty} \int_{n}^{n+1} f(x) dx$$ #Step 3: Write the sum using the integral#
02

From the inequality we derived in Step 2, we can now write the sum as an integral: $$\sum_{n=1}^{\infty} a_n \leq a_1 + \int_{1}^{\infty} f(x) dx$$ #Conclusion#

Since we have shown that \(\sum_{n=1}^{\infty} a_{n} \leq a_{1}+\int_{1}^{\infty} f(x) dx\), the statement is true. The sum of the series is less than or equal to the sum of the area under the curve of the function \(f(x)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Maria sees the growth of her business for the upcoming year as being tied to the gross domestic product (GDP). She believes that her business will grow (or contract) at the rate of \(5 \%, 4.5 \%, 3 \%, 0 \%\), or \(-0.5 \%\) per year if the GDP grows (or contracts) at the rate of between 2 and \(2.5 \%\), between \(1.5\) and \(2 \%\), between 1 and \(1.5 \%\), between 0 and \(1 \%\), and between \(-1\) and \(0 \%\), respectively. Maria has decided to assign a probability of \(.12, .24, .40, .20\), and \(.04\), respectively, to each outcome. At what rate does Maria expect her business to grow next year?

A survey of 1000 families was conducted by the Public Housing Authority in a certain community to determine the distribution of families by size. The results follow: $$\begin{array}{lccccccc}\hline \text { Family Size } & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text { Frequency of } & & & & & & & \\ text { Occurrence } & 350 & 200 & 245 & 125 & 66 & 10 & 4 \\ \hline\end{array}$$ a. Find the probability distribution of the random variable \(X\), where \(X\) denotes the number of persons in a randomly chosen family. b. Draw the histogram corresponding to the probability distribution found in part (a).

The number of average hours worked per year per worker in the United States and five European countries in 2002 is given in the following table: $$\begin{array}{lcccccc}\hline & & & \text { Great } & & \text { West } & \\\\\text { Country } & \text { U.S. } & \text { Spain } & \text { Britain } & \text { France } & \text { Germany } & \text {Norway } \\ \hline \text { Average } & & & & & & \\\\\text { Hours } & 1815 & 1807 & 1707 & 1545 & 1428 & 1342 \\ \text { Werked } & & & & & & \\ \hline\end{array}$$ Find the average of the average hours worked per worker in 2002 for workers in the six countries. What is the standard deviation for these data?

In a lottery, 5000 tickets are sold for $$\$ 1$$ each. One first prize of $$\$ 2000,1$$ second prize of $$\$ 500,3$$ third prizes of $$\$ 100$$, and 10 consolation prizes of \(\$ 25\) are to be awarded. What are the expected net earnings of a person who buys one ticket?

Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The number of hours a child watches television on a given day
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.