Problem 52 - College Algebra Essentials

Question

Find the partial fraction decomposition for \(\frac{2}{x(x+2)}\) and use the result to find the following sum: $$ \frac{2}{1 \cdot 3}+\frac{2}{3 \cdot 5}+\frac{2}{5 \cdot 7}+\dots+\frac{2}{99 \cdot 101} $$

Step-by-Step Solution

Answer

The sum \(\frac{2}{1 \cdot 3}+\frac{2}{3 \cdot 5}+\frac{2}{5 \cdot 7}+\dots+\frac{2}{99 \cdot 101}\) equals \(1 - \frac{1}{101}\).

Step 1: Performing Partial Fraction Decomposition

Start with \(\frac{2}{x(x+2)}\). Suppose, \(\frac{2}{x(x+2)} = \frac{A}{x} + \frac{B}{x+2}\). Multiplying throughout by \(x(x+2)\), we get \(2 = A(x+2) + Bx\). Solving this linear system of equations, we obtain A = 1 and B = -1. So, \(\frac{2}{x(x+2)} = \frac{1}{x} - \frac{1}{x+2}\) after decomposition.

Step 2: Identify Series

Observing the sum, \(\frac{2}{1 \cdot 3}+\frac{2}{3 \cdot 5}+\frac{2}{5 \cdot 7}+\dots+\frac{2}{99 \cdot 101}\), each term can be written as: \(\frac{2}{n(n+2)} = \frac{1}{n} - \frac{1}{n+2}\), matching fully with the decomposed form derived from Step 1.

Step 3: Evaluate the Sum

The sum: \(\frac{2}{1 \cdot 3}+\frac{2}{3 \cdot 5}+\frac{2}{5 \cdot 7}+\dots+\frac{2}{99 \cdot 101}\) transforms into: \(\left(1-\frac{1}{3}\right) + \left(\frac{1}{3}-\frac{1}{5}\right) + \left(\frac{1}{5}-\frac{1}{7}\right)+...+ \left(\frac{1}{99}-\frac{1}{101}\right)\). Most terms cancel out with others, just leaving \(1 - \frac{1}{101}\)

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating those symbols. It's essential for solving problems, especially when working with equations and fractions.
The main idea is to represent numbers with variables like \(x\), \(y\), and \(z\), and to use algebraic expressions to describe quantities that change according to defined patterns.
In partial fraction decomposition, algebra allows us to break down complex fractions into simpler parts. For instance, given \(\frac{2}{x(x+2)}\), we can express it as a sum of simpler fractions \(\frac{A}{x} + \frac{B}{x+2}\).
Here are some key steps:

• Assume a form for the decomposition, using constants \(A\) and \(B\).
• Multiply through by the denominator to clear fractions.
• Use algebraic manipulation to solve for \(A\) and \(B\).

Solving these systems of equations is foundational in algebra and helps deduce simpler forms for complex rational expressions, making further calculations or evaluations more straightforward.

Series

A series is the sum of the terms of a sequence. It's a way to represent the total of a sequence of numbers, whether finite or infinite.
With arithmetic series, each term is obtained by adding a constant to the previous term. In our problem, we identify the series based on the decomposed form of partial fractions.
Our series:

• Represents a telescoping pattern where many terms cancel out.
• Starts with terms \(\frac{2}{1 \cdot 3}, \frac{2}{3 \cdot 5}, \frac{2}{5 \cdot 7}\), etc.
• Uses transformation with the help of partial fraction decomposition \(\frac{1}{n} - \frac{1}{n+2}\).

In analyzing such series, the focus is usually on the first and last terms, as intermediate terms cancel out. This property makes complex sum calculations manageable and shows the elegance of series manipulation.

Sum of Fractions

Adding fractions involves finding a common denominator. Yet, partial fractions allow an easier approach when summed as part of a series, particularly in a telescoping series.
Let's consider the example series from the exercise, in function of the partial fractions:

• The series simplifies due to cancellation of many middle terms.
• Initially, evaluate terms like \(1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5}\).
• Continues this pattern through to \(\frac{1}{99} - \frac{1}{101}\).

Due to telescoping, most intermediate fractions disappear, and the sum ultimately simplifies to the first term minus the last term, in this case, \(1 - \frac{1}{101}\).
This demonstrates how organizing fractions leveraging their patterns and transformations results in efficient sum calculations, reinforcing the importance of mathematical strategies like partial fraction decomposition.