Problem 42 - Precalculus

Question

Find the partial-fraction decomposition of \(\frac{1}{n(n+1)}\) and apply it to find the sum of $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{999 \cdot 1000}$$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{999}{1000} \).

Step 1: Identify the Partial-Fraction Form

We want to express \( \frac{1}{n(n+1)} \) as a sum of two simpler fractions. Set up the equation \( \frac{1}{n(n+1)} = \frac{A}{n} + \frac{B}{n+1} \).

Step 2: Clear the Denominator

Multiply both sides by \( n(n+1) \) to get rid of the denominators on the right-hand side: \( 1 = A(n+1) + Bn \).

Step 3: Expand and Simplify

Expand the equation: \( 1 = An + A + Bn \) which simplifies to \( 1 = (A + B)n + A \).

Step 4: Solve for Coefficients

Compare coefficients on both sides: For the \( n \) terms, \( A + B = 0 \). For the constant term, \( A = 1 \). Solving these equations, we obtain \( A = 1 \) and \( B = -1 \).

Step 5: Express the Partial-Fractions

Substitute \( A \) and \( B \) back into the equation: \( \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \).

Step 6: Identify the Sum

To find the sum \( \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{999 \cdot 1000} \), substitute the partial-fraction decomposition into each term.

Step 7: Write Out the Series

Write the series based on the decomposition: \( \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \ldots + \left( \frac{1}{999} - \frac{1}{1000} \right) \).

Step 8: Observe the Telescoping Series

Notice that in the expanded form most intermediate terms cancel out: \( 1 - \frac{1}{1000} \).

Step 9: Calculate the Remaining Terms

The only terms that do not get canceled are the first and the last: \( 1 - \frac{1}{1000} = \frac{1000}{1000} - \frac{1}{1000} = \frac{999}{1000} \).

Key Concepts

Telescoping Series

A telescoping series is a special type of series where consecutive terms cancel each other out, leaving only a few terms at the beginning and end of the series. This cancellation makes it much easier to determine the sum of the series. In our exercise, each term in the series was decomposed into

• \( \frac{1}{n} \)
• \( - \frac{1}{n+1} \)

For example, for the series:
\[ \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \ldots + \left( \frac{1}{999} - \frac{1}{1000} \right) \]
many terms cancel each other, leading to a very reduced form. The beauty of a telescoping series is how simple and elegant it is to calculate the final sum once you recognize the cancellation pattern. For the provided example, the sum reduces to:
\[ 1 - \frac{1}{1000} = \frac{999}{1000} \]

Precalculus

Precalculus is a branch of mathematics that prepares students for calculus. It covers various foundational concepts necessary for understanding calculus. One of the central topics in precalculus is the study of sequences and series, including techniques like partial fraction decomposition, which is a necessary skill when working with complex algebraic expressions.Understanding how to decompose a rational expression into partial fractions can simplify the process of solving integrals or sums. Many problems in precalculus begin with what might seem like complicated fractions, similar to the expression \( \frac{1}{n(n+1)} \) in our exercise.
By learning to express it as \( \frac{1}{n} - \frac{1}{n+1} \), you acquire techniques useful in more advanced calculus topics. Precalculus ensures you are comfortable with these kinds of transformations and can see their practical use in mathematical problem-solving.

Rational Expressions

Rational expressions are fractions composed of polynomials. They are an essential part of algebra and precalculus as they appear frequently in mathematical problems and real-life applications. In this exercise, we examined the expression
\[ \frac{1}{n(n+1)} \]
which is a particular rational expression.The goal of partial fraction decomposition in this context is to break down a complex rational expression into simpler, easily manageable fractions. This process involves determining coefficients that satisfy original constraints, allowing the expression to be rewritten in a simpler form.

• Set up the equation for the rational expression \( \frac{1}{n(n+1)} \)
• Clear the denominators and expand appropriately
• Solve for the constants to achieve the decomposition

The method of decomposition makes it straightforward to evaluate expressions, especially in series, by simplifying each fraction individually before proceeding to find sums or integrals. Mastery over rational expressions is vital for mathematical problem-solving in advanced topics.