Problem 68 - Precalculus

Question

In calculus, partial fractions are used to calculate the sums of infinite series. Find the partial-fraction decomposition of the summand. $$\sum_{k=1}^{\infty} \frac{4}{k(k+1)(k+2)}$$

Step-by-Step Solution

Answer

The partial-fraction decomposition of \( \frac{4}{k(k+1)(k+2)} \) is \( \frac{2}{k} - \frac{4}{k+1} + \frac{2}{k+2} \).

Step 1: Set Up the Partial Fraction Decomposition

To decompose \( \frac{4}{k(k+1)(k+2)} \) into partial fractions, assume \( \frac{4}{k(k+1)(k+2)} = \frac{A}{k} + \frac{B}{k+1} + \frac{C}{k+2} \). This setup allows us to solve for the constants \(A\), \(B\), and \(C\) by equating coefficients.

Step 2: Clear the Denominator

Multiply both sides of the equation by the denominator \(k(k+1)(k+2)\) to get rid of the fractions: \[ 4 = A(k+1)(k+2) + Bk(k+2) + Ck(k+1) \]

Step 3: Expand and Simplify

Expand the right-hand side of the equation: \[ A(k^2 + 3k + 2) + B(k^2 + 2k) + C(k^2 + k) = 4 \] Combine the terms: \[ (A + B + C)k^2 + (3A + 2B + C)k + 2A = 4 \] Equate this to \(4\), which means \((A + B + C)k^2 + (3A + 2B + C)k + 2A = 0k^2 + 0k + 4\).

Step 4: Solve for Constants

By comparing coefficients, set up the system of equations: 1. \(A + B + C = 0\) 2. \(3A + 2B + C = 0\) 3. \(2A = 4\) Solve these equations: From equation 3, \(A = 2\). Substituting \(A\) into equations 1 and 2 gives:- Equation 1: \(2 + B + C = 0\), so \(B + C = -2\) - Equation 2: \(6 + 2B + C = 0\), so \(2B + C = -6\) Solve for \(B\) and \(C\): Subtract equation for \(B + C\) from equation for \(2B + C\): \((2B + C) - (B + C) = -6 - (-2)\) \(B = -4\) Substitute \(B = -4\) into \(B + C = -2\): \(-4 + C = -2\) \(C = 2\). Thus, \(A = 2\), \(B = -4\), \(C = 2\).

Step 5: Write the Partial Fraction Decomposition

Substituting \(A\), \(B\), and \(C\) back, the partial fraction decomposition is: \[ \frac{4}{k(k+1)(k+2)} = \frac{2}{k} - \frac{4}{k+1} + \frac{2}{k+2} \]

Key Concepts

Infinite Series

Infinite series are sums of infinite sequences of numbers. They appear frequently in calculus and help describe functions that cannot be simplified into a finite form. Understanding how to work with infinite series is key to tackles more complex mathematical concepts, such as convergence and divergence.
This exercise requires finding a partial fraction decomposition related to an infinite series, enabling us to determine what happens to the series as we sum more and more terms.

• Infinite series are often associated with sequences such as geometric series or power series.
• The general idea is to sum up terms indefinitely, and determine whether this sum reaches a finite value.

In practical terms, calculating the sum of infinite series often involves transforming the expression using methods like partial fraction decomposition, which simplifies complicated fractions into a sum of simpler ones.
By understanding infinite series, particularly their convergence, we can analyze how functions behave over an interval or as a variable approaches infinity, which is a crucial skill in calculus.

Calculus

Calculus is a branch of mathematics that explores changes through the use of derivatives and integrals. It is widely used in science and engineering for modeling situations where variables change with respect to one another or to time.
Within calculus, one common application is analyzing series, like infinite series, for convergence—determining if they sum up to a finite value.

• Calculus employs methods such as integration and differentiation to solve problems involving motion, growth, etc.
• Partial fraction decomposition simplifies functions, making them easier to integrate or differentiate.

The problem of decomposing a rational function into simpler components using partial fractions is an excellent application of algebraic manipulation within calculus. Calculus techniques, like integration of such decomposed functions, allow us to find areas, volumes, and other quantities that describe the physical world.
The step-by-step approach involves looking at the powers of polynomials and their simplification, which shows the underlying principles of calculus acting on otherwise complex expressions.

System of Equations

Systems of equations are collections of two or more equations with a common set of variables. They are powerful in both algebra and calculus for finding points of intersection, optimizing functions, and more.
In the context of partial fraction decomposition, solving a system of equations allows us to find coefficients that break down a rational expression into simpler terms.

• Systems can be solved using various methods, including substitution, elimination, and matrices.
• Simplification into a system lets us handle complex algebraic expressions systematically.

In the exercise, the system of equations was derived from equating polynomial coefficients on both sides of a fraction identity. Solving these allows us to find the constants needed to express a fraction as a sum of simpler parts.
This approach not only shows the power of algebra in solving calculus problems but also demonstrates the systematic nature of mathematics, where one method builds upon another to reach a solution.