Problem 45
Question
Find the sum of each series in Exercises \(45-52 .\) $$\sum_{n=1}^{\infty} \frac{4}{(4 n-3)(4 n+1)}$$
Step-by-Step Solution
Verified Answer
The sum of the series is 1.
1Step 1: Identify the series
The given series is \( \sum_{n=1}^{\infty} \frac{4}{(4n-3)(4n+1)} \). This is an infinite series where each term is given by \( \frac{4}{(4n-3)(4n+1)} \).
2Step 2: Partial Fraction Decomposition
To simplify the series, express the term \( \frac{4}{(4n-3)(4n+1)} \) using partial fractions. We write \( \frac{4}{(4n-3)(4n+1)} = \frac{A}{4n-3} + \frac{B}{4n+1} \) and solve for A and B.
3Step 3: Solve for Constants A and B
Multiply both sides by \((4n-3)(4n+1)\) to get \(4 = A(4n+1) + B(4n-3)\). Expand and combine terms: \((4A + 4B)n + (A - 3B) = 4\). This gives two equations: \(4A + 4B = 0\) and \(A - 3B = 4\). Solve these equations to find \(A = 1\) and \(B = -1\).
4Step 4: Rewrite the Series
Substitute \(A\) and \(B\) back into the partial fraction form: \( \frac{4}{(4n-3)(4n+1)} = \frac{1}{4n-3} - \frac{1}{4n+1} \). The series \( \sum_{n=1}^{\infty} \left( \frac{1}{4n-3} - \frac{1}{4n+1} \right) \) is now a telescoping series.
5Step 5: Evaluate the Telescoping Series
Write out the first few terms of the expanded series to observe the cancellation pattern. The first few terms are \( \frac{1}{1} - \frac{1}{5} + \frac{1}{5} - \frac{1}{9} + \frac{1}{9} - \frac{1}{13} + \ldots \). Most terms cancel each other, leaving only the first term from the series: \( \frac{1}{1} \).
6Step 6: Conclude the Sum
Since the series is telescoping and all terms cancel except the initial segment, the sum of the series is the remaining term: \(1\).
Key Concepts
Partial Fraction DecompositionInfinite SeriesSeries ConvergenceMathematical Proofs
Partial Fraction Decomposition
Partial fraction decomposition is a method used to break down complex rational expressions into simpler ones. This technique is especially useful when dealing with complex fractions that can be broken into more manageable parts.
To perform partial fraction decomposition, we express a given fraction as a sum of simpler fractions. In our exercise, we simplify \( \frac{4}{(4n-3)(4n+1)} \) into \( \frac{A}{4n-3} + \frac{B}{4n+1} \).
This step is crucial in solving summation problems involving series, as it allows us to transform a difficult expression into smaller terms that can be easily managed. Solving for constants \(A\) and \(B\) involves creating and solving a system of equations by equating the coefficients for corresponding powers of the variable.
To perform partial fraction decomposition, we express a given fraction as a sum of simpler fractions. In our exercise, we simplify \( \frac{4}{(4n-3)(4n+1)} \) into \( \frac{A}{4n-3} + \frac{B}{4n+1} \).
This step is crucial in solving summation problems involving series, as it allows us to transform a difficult expression into smaller terms that can be easily managed. Solving for constants \(A\) and \(B\) involves creating and solving a system of equations by equating the coefficients for corresponding powers of the variable.
Infinite Series
An infinite series is the sum of an infinite sequence of numbers, often represented as \( \sum_{n=1}^{fty} a_n \). The concept of infinite series is essential in calculus and mathematical analysis.
In our problem, the series is expressed as \( \sum_{n=1}^{fty} \frac{4}{(4n-3)(4n+1)} \). Infinite series can converg, diverge, or oscillate, depending on the terms and their behavior as \( n \) becomes very large.
This type of series often appears in mathematical problems where a sequence's sum trends toward a specific value, infinity, or varies without settling into any pattern. Understanding how these series work is key to mathematical studies that involve calculations of accumulated quantities, like Physics or Economics.
In our problem, the series is expressed as \( \sum_{n=1}^{fty} \frac{4}{(4n-3)(4n+1)} \). Infinite series can converg, diverge, or oscillate, depending on the terms and their behavior as \( n \) becomes very large.
This type of series often appears in mathematical problems where a sequence's sum trends toward a specific value, infinity, or varies without settling into any pattern. Understanding how these series work is key to mathematical studies that involve calculations of accumulated quantities, like Physics or Economics.
Series Convergence
Series convergence is a property of an infinite series where the sum of the terms approaches a finite limit as more terms are added. It's important in identifying if a series has a specific sum or if it diverges.
To check for convergence, various tests can be used, such as the ratio test, root test, and comparison test. For our exercise, the series itself is telescoping, which makes it easier to determine convergence.
In a telescoping series, cancelation of terms helps in seeing convergence visually. In this case, the series \( \sum_{n=1}^{fty} \left( \frac{1}{4n-3} - \frac{1}{4n+1} \right) \) has terms that cancel out sequentially, leaving only a few remaining terms that add to a known value, indicating convergence.
To check for convergence, various tests can be used, such as the ratio test, root test, and comparison test. For our exercise, the series itself is telescoping, which makes it easier to determine convergence.
In a telescoping series, cancelation of terms helps in seeing convergence visually. In this case, the series \( \sum_{n=1}^{fty} \left( \frac{1}{4n-3} - \frac{1}{4n+1} \right) \) has terms that cancel out sequentially, leaving only a few remaining terms that add to a known value, indicating convergence.
Mathematical Proofs
Mathematical proofs are logical arguments that establish the truth of a mathematical statement. They are vital for validating results in mathematics.
For our infinite series, we prove convergence by employing the partial fraction technique, which leads to a telescoping series. The proof involves simplifying the problem, often after employing strategic methods like partial fraction decomposition, to reveal the nature of the series.
This telescoping nature is the core of the proof. The cancellation pattern shows that after all intermediate terms vanish, we are left with a finite sum, in this case, \(1\). The remaining single term is the sum of the entire series, with the pattern providing visual and mathematical certainty of convergence.
For our infinite series, we prove convergence by employing the partial fraction technique, which leads to a telescoping series. The proof involves simplifying the problem, often after employing strategic methods like partial fraction decomposition, to reveal the nature of the series.
This telescoping nature is the core of the proof. The cancellation pattern shows that after all intermediate terms vanish, we are left with a finite sum, in this case, \(1\). The remaining single term is the sum of the entire series, with the pattern providing visual and mathematical certainty of convergence.
Other exercises in this chapter
Problem 45
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