Question

$u_{x}=\frac{1}{x(x+2)} ; \quad x=1,2,3, \ldots$

   $u_{x}=\frac{1}{x(x+2)} ; \quad x=1,2,3, \ldots$

The Chemistry Maths Book

The Chemistry Maths Book

Erich Steiner 2nd Edition

Chapter 7, Problem 6

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Solved by Expert John Nicolle

01/01/2022

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We need to find the first six terms of this sequence. Show more…

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$u_{x}=\frac{1}{x(x+2)} ; \quad x=1,2,3, \ldots$

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Key Concepts

Telescoping Series

A telescoping series is a series in which many terms cancel out when the series is expanded. When a sequence defined by partial fractions is summed, the resulting series often exhibits cancellation of intermediate terms, leading to a much simpler expression for the sum. Recognizing and exploiting this telescoping behavior is a powerful technique in evaluating series that might initially appear complex.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is especially useful when the rational function’s denominator can be factored into linear or irreducible quadratic factors. In the given sequence, decomposing 1/(x(x+2)) into partial fractions can simplify further analysis, such as summing series or integrating the function, by breaking it down into simpler terms.

Sequence

A sequence is an ordered list of numbers defined by a specific formula or rule. In this context, the sequence is defined by u(x) = 1/(x(x+2)) for natural numbers x. Sequences are fundamental in mathematics as they provide a framework for studying the behavior of ordered sets of numbers, whether that involves convergence, divergence, or other properties in analysis.

Rational Function

A rational function is a ratio of two polynomials. In the provided sequence, the term u(x) is expressed as 1/(x(x+2)), which is a particular type of rational function. Understanding rational functions is important because they appear frequently in problems involving sequences, series, and calculus, and their properties can be exploited to simplify or analyze mathematical expressions.

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