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Chapter 5: 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 given sequence is \( \frac{99}{100} \)

Step by step solution

01

Perform the Partial Fraction Decomposition

It is known that any rational function can be written as a sum of simpler fractions, a process called partial fraction decomposition. Given the fraction \( \frac{1}{x(x+1)} \), this can be represented as \( \frac{A}{x} + \frac{B}{x+1} \) for constants A and B. By setting \( x(x+1) [ \frac{A}{x} + \frac{B}{x+1} ] = 1 \), we end up with the equation \( A(x+1) + Bx = 1 \). Plugging in \(x = 0\) results in \( A = 1 \). Plugging in \(x = -1\) results in \( B = -1 \). Therefore, the partial fraction decomposition of \( \frac{1}{x(x+1)} \) is \( \frac{1}{x} - \frac{1}{x+1} \).
02

Analyze the Sequence

The sequence from the problem, \( \frac{1}{1 \cdot 2}, \frac{1}{2 \cdot 3}, \frac{1}{3 \cdot 4},..., \frac{1}{99 \cdot 100} \) can be rewritten using the partial fractions found in Step 1. Here, \( \frac{1}{1 \cdot 2} = \frac{1}{1} - \frac{1}{2},\frac{1}{2 \cdot 3} = \frac{1}{2} - \frac{1}{3},\frac{1}{3 \cdot 4} = \frac{1}{3} - \frac{1}{4},..., \frac{1}{99 \cdot 100} = \frac{1}{99} - \frac{1}{100} \)
03

Add the Sequence

By adding all the terms of the sequence together, all of the negative fractions from one term will cancel with the positive fraction of the preceding term, leaving only \( 1 - \frac{1}{100} = \frac{99}{100} \)

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Most popular questions from this chapter

In Exercises \(5-14,\) an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$\begin{aligned}&z-4 x+2 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\2 x+3 y \leq 12\end{array}\right.\\\&\begin{array}{l}3 x+2 y \leq 12 \\\x+y \geq 2\end{array}\end{aligned}$$

What is a system of linear inequalities?

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is \(\$ 125\) for the rear-projection televisions and \(\$ 200\) for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and let \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit. b. The manufacturer is bound by the following constraints: \(\cdot\) Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \(\cdot\) Equipment in the factory allows for making at most 200 plasma televisions in one month. \(\cdot\) The cost to the manufacturer per unit is \(\$ 600\) for the rear- projection televisions and \(\$ 900\) for the plasma televisions. Total monthly costs cannot exceed \(\$ 360,000\). Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200),(450,100),\) and \((450,0) .]\) e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing- rear-projection televisions each month and maximum monthly profit is $\$$

In Exercises 106-109, determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing \(3 x-4 y<12,\) it's not necessary for me to graph the linear equation \(3 x-4 y-12\) because the inequality contains a \(<\) symbol, in which equality is not included.

Solve the systems $$\left\\{\begin{array}{l} \log x^{2}-y+3 \\ \log x-y-1 \end{array}\right.$$
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