/*! 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 51 Find the partial fraction decomp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 51

Find the partial fraction decomposition for \(\frac{1}{x(x+1)}\) and use the result to find the following sum: $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{99 \cdot 100}$$

Short Answer

Expert verified
The sum of the series is \( \frac{99}{100} \).

Step by step solution

01

Find the partial fraction decomposition

We can express the given fraction, \( \frac{1}{x(x+1)} \), as the sum of two simpler fractions: \( \frac{A}{x} \) and \( \frac{B}{x+1} \), for some coefficients A and B: \n\n\( \frac{1}{x(x+1)} = \frac{A}{x} + \frac{B}{x+1} \)\n\nMultiplying throughout by the common denominator, \( x(x+1) \), gives us: \n\n\( 1 = A(x+1) + Bx \)\n\nBy comparing coefficients, we can determine A and B. Setting x = 0 gives A = 1, and setting x = -1 gives B = -1. So, our partial fraction decomposition is: \n\n\( \frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x+1} \)
02

Express the given sum using partial fractions

Using the partial fraction decomposition we've found, we can express each term of the given sum, \( \frac{1}{n(n+1)} \), as the difference between two simpler fractions: \n\n\( \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \)\n\nSo, the given sum can be written as: \n\n\( \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+ \dots+\frac{1}{99 \cdot 100} = (1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}- \frac{1}{4})+ \dots+(\frac{1}{99}-\frac{1}{100}) \)
03

Calculate the sum

Now observe that in the sum obtained in the previous step, every term from \( \frac{1}{2} \) to \( \frac{1}{99} \) appears twice and gets cancelled out. We only have the first and the last term remaining: \n\n\( 1 - \frac{1}{100} \)\n\nSo, the sum of the given series is \( \frac{99}{100} \).

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

Solve the systems. $$ \left\\{\begin{array}{l} {\log _{y} x=3} \\ {\log _{y}(4 x)=5} \end{array}\right. $$

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. In 1978 , a ruling by the Civil Aeronautics Board allowed Federal Express to purchase larger aircraft. Federal Express's options included 20 Boeing 727 s that United Airlines was retiring and/or the French-built Dassault Fanjet Falcon \(20 .\) To aid in their decision, executives at Federal Express analyzed the following data: $$\begin{array}{ll} {} & {\text { Boeing } 727 \quad \text { Falcon } 20} \\ {\text { Direct Operating cost }} & {\$ 1400 \text { per hour } \$ 500 \text { per hour }} \\ {\text { Payload }} & {42,000 \text { pounds } \quad 6000 \text { pounds }} \end{array}$$ Federal Express was faced with the following constraints: \(\cdot\) Hourly operating cost was limited to 35,000. \(\cdot\) Total payload had to be at least 672,000 pounds. \(\cdot\) Only twenty 727 s were available. Given the constraints, how many of each kind of aircraft should Federal Express have purchased to maximize the number of aircraft?

Exercises \(41-43\) will help you prepare for the material covered in the first section of the next chapter. Solve the system: $$ \left\\{\begin{aligned} w-x+2 y-2 z &=-1 \\ x-\frac{1}{3} y+z &=\frac{8}{3} \\ y-z &=1 \\ z &=3 \end{aligned}\right. $$ Express the solution set in the form {(w, x, y, z)\\} . What makes it fairly easy to find the solution?

determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \((x+3)^{2}\) consists of two factors of \(x+3,1\) set up the following partial fraction decomposition: $$\frac{5 x+2}{(x+3)^{2}}=\frac{A}{x+3}+\frac{B}{x+3}$$

When a crew rows with the current, it travels 16 miles in 2 hours. Against the current, the crew rows 8 miles in 2 hours. Let \(x=\) the crew's rowing rate in still water and let \(y=\) the rate of the current. The following chart summarizes this information:
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.