Question

Consider the sequence \(\left\\{F_{n}\right\\}\) defined by $$F_{n}=\sum_{k=1}^{\infty} \frac{1}{k(k+n)^{\prime}}$$ for \(n=0,1,2, \ldots . .\) When \(n=0,\) the series is a \(p\) -series, and we have \(F_{0}=\pi^{2} / 6\) (Exercises 65 and 66 ). for \(n=0,1,2, \ldots . .\) When \(n=0,\) the series is a \(p\) -series, and we have \(F_{0}=\pi^{2} / 6\) (Exercises 65 and 66 ).

Short answer

Question: Write the formula for finding the value of \(F_n\) for any positive integer \(n\). Answer: The formula for finding the value of \(F_n\) for any positive integer \(n\) is \(F_n = \frac{1}{n} \left(\frac{\pi^2}{6} - \sum_{k=1}^{\infty} \frac{1}{k+n}\right)\).

Step-by-step solution

Simplify the expression for the sequence's term \(F_n\)

Using partial fraction decomposition, rewrite the expression \(\frac{1}{k(k+n)}\) as the difference of two simpler fractions with simpler denominators: $$\frac{1}{k(k + n)} = \frac{A}{k} + \frac{B}{k + n}$$ To find \(A\) and \(B\), clear the denominators by multiplying both sides by \(k(k+n)\): $$1 = A(k+n) + Bk$$ Now, we will find the values of A and B by choosing specific values for k.

Determine the values of A and B

Let's choose \(k=0\), we get: $$1 = A(0+n)+B(0) \Rightarrow 1 = An \Rightarrow A = \frac{1}{n}$$ Now, let's choose \(k=-n\), we get: $$1 = A(-n+n)+B(-n) \Rightarrow 1 = -Bn \Rightarrow B = -\frac{1}{n}$$ Now we can rewrite the original expression for \(F_n\) as: $$F_n = \sum_{k=1}^{\infty} \left(\frac{1}{n}\frac{1}{k} - \frac{1}{n}\frac{1}{k+n}\right)$$

Distribute the sum notation

Distribute the sum over the expression inside the parentheses: $$F_n = \frac{1}{n}\sum_{k=1}^{\infty} \frac{1}{k} - \frac{1}{n}\sum_{k=1}^{\infty} \frac{1}{k+n}$$

Recognize the p-series in the expressions

The two sums above are both p-series. We know that for \(n=0\), \(F_0 = \pi^2/6\). According to the given problem, we obtain the sum of the harmonic series as \(F_0 = \pi^2/6\). Therefore, we can substitute the value into the expression: $$F_n = \frac{1}{n} \left(\frac{\pi^2}{6} - \sum_{k=1}^{\infty} \frac{1}{k+n}\right)$$ This formula allows us to find the values of \(F_n\) for any positive integer \(n\).

Key concepts

P-Series

The term "p-series" refers to a particular type of series where each term is of the form \( \frac{1}{k^p} \), where \( p \) is a positive constant and \( k \) is an integer starting from 1. These series are an important class of mathematical series, often examined for their convergence properties.

The most famous p-series is the harmonic series, which occurs when \( p = 1 \). In general, a p-series converges if \( p > 1 \) and diverges if \( p \leq 1 \).

For example:

• If \( p = 2 \), we have \( \sum_{k=1}^{\infty} \frac{1}{k^2} \), which is known to converge. In fact, it converges to \( \frac{\pi^2}{6} \).
• If \( p = 1 \), the series becomes the harmonic series, which diverges.

Understanding p-series is crucial for series analysis as it sets a foundational pattern for determining the behavior of many infinite series.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex fractions into a sum of simpler fractions. This is particularly helpful when evaluating series and integrals, as it allows us to apply known techniques to solve simpler parts.

For a fraction like \( \frac{1}{k(k+n)} \), partial fraction decomposition lets us write:

• \( \frac{1}{k(k+n)} = \frac{A}{k} + \frac{B}{k+n} \)

Here, \( A \) and \( B \) are constants found by solving the equation obtained from clearing the fractions. In our exercise, this helps simplify the computation of \( F_n \), making it easier to recognize the pattern and apply knowledge of series convergence.

This process involves finding values for \( A \) and \( B \) by assigning strategic values to \( k \) and solving resulting equations, thus transforming the original expression into the sum of simpler components.

Harmonic Series

The harmonic series is an important series where each term is \( \frac{1}{k} \). It is defined as:

\( \sum_{k=1}^{\infty} \frac{1}{k} \)

This series is well-known for its divergence, despite the terms approaching zero as \( k \) increases. The divergence of the harmonic series is an essential concept in mathematical analysis as it defies the intuition that smaller terms should make a series converge.

Understanding this helps when evaluating many series and determining their convergence properties. Even though \( \frac{1}{k} \) becomes very small, the sum continues to grow without bound, thus making it clear that smaller terms do not always guarantee convergence.

In our sequence problem, the harmonic series is embedded in one of the parts of the sum, demonstrating the need for careful manipulation and evaluation when dealing with such series.

Sequence

A sequence is a list of numbers arranged in a specific order, often generated by a formula or rule. In our exercise, the sequence \( \{ F_n \} \) is defined using a summation formula.

Each term \( F_n \) in this sequence is calculated by an infinite series, particularly highlighting series convergence and manipulation techniques like partial fraction decomposition. Sequences can showcase behavior patterns such as increasing, decreasing, or oscillating values.

Working with sequences is pivotal in mathematics as they lay the groundwork for more complex constructs like series and mathematical functions. Understanding sequences allows you to explore conditions of convergence and divergence, vital for dealing with infinite processes.

• For example, the sequence \( \{ F_n \} \) has terms represented by a complex series formula that requires careful computational techniques to evaluate and understand behavior across different values of \( n \).

Mastery over sequences allows one to analyze the performance and outcome of expressions over an extended range of numbers.