Question

Consider the sequence $\text{Missing left or extra right}$ defined by $$F_n=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k(k+n)},$$ for $n=0,1,2,\dots ..$ When $n=0,$ the series is a $p$ -series, and we have $F_0={\pi }^2/6$ (Exercises 65 and 66 ). a. Explain why $\text{Missing left or extra right}$ is a decreasing sequence. b. Plot $\text{Missing left or extra right}$ for $n=1,2,\dots ,20$. c. Based on your experiments, make a conjecture about $\underset{n\to \mathrm{\infty }}{lim}{F}_n$.

Short answer

Answer: Yes, the sequence $\text{Missing left or extra right}$ is a decreasing sequence, since $F_{n+1}<F_n$ for all values of n.

Step-by-step solution

Proving that the Sequence is Decreasing

To prove that the sequence is decreasing, we will need to show that $F_{n+1}<F_n$ for every n. Considering the partial sums of the series: $F_n=\frac{1}{1(1+n)}+\frac{1}{2(2+n)}+\frac{1}{3(3+n)}+\cdots$, and $F_{n+1}=\frac{1}{1(1+n+1)}+\frac{1}{2(2+n+1)}+\frac{1}{3(3+n+1)}+\cdots$. Now, let's compare the individual terms of the two partial sums: $\frac{1}{1(1+n+1)}<\frac{1}{1(1+n)}$, $\frac{1}{2(2+n+1)}<\frac{1}{2(2+n)}$, $\frac{1}{3(3+n+1)}<\frac{1}{3(3+n)}$, and so on. Since all the terms in the partial sum representation of $F_{n+1}$ are less than the corresponding terms in the partial sum representation of $F_n$, we can conclude that $F_{n+1}<F_n$ for all n. Hence, the sequence $\text{Missing left or extra right}$ is decreasing.

Calculate Components of the Sequence

To plot the sequence, we first need to calculate the first 20 components of the sequence. For these small values of n, we can obtain an approximate value of each component by calculating the sum up to a certain minimum number of terms such that the error from truncating the infinite sum is negligible. We can use a computer algebra system such as Mathematica or a powerful calculator to get the approximate values for $F_1,F_2,\dots ,F_{20}$.

Plot the Sequence

After calculating the first 20 components of the sequence, we can plot these values on a graph. You can use any graphing software, or a graphing calculator to accomplish this. The graph should show a clear decreasing trend in the sequence.

Conjecturing the Limit

Based on the decreasing trend of the sequence and the calculated values of the components, we can make a conjecture about the limit of the sequence as n approaches infinity. Since the terms are decreasing and positive, it is reasonable to conjecture that: $\underset{n\to \mathrm{\infty }}{lim}{F}_n=0$. This conjecture will need to be proved rigorously in a more advanced mathematical setting, but based on the graphical plot, our conjecture seems reasonable for the given problem.

Key concepts

P-Series

A p-series is a type of infinite series given by the general form $\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^p}$ for a constant p.

To determine whether a p-series converges, one must look at the value of p. If p is greater than 1, the series converges, due to the integral test for convergence. On the other hand, if p is less than or equal to 1, the series diverges. The sequence in the given exercise represents a p-series when n=0, and it is known to converge to ${\pi }^2/6$ since p would be 2 in that case.

Series Comparison Test

The Series Comparison Test is a method used to determine the convergence or divergence of an infinite series by comparing it with another series whose convergence is already known.

• If the series in question is less than or equal to a convergent series term-by-term, then it must also converge.
• If it is greater than or equal to a divergent series term-by-term, it must also diverge.

In our exercise, you could use the comparison test to compare the given series with a p-series, ensuring each term of our series is smaller than the corresponding terms of the converging p-series when n=0, therefore implying the given series also converges.

Mathematical Conjecture

A mathematical conjecture is an educated guess or hypothesis about a pattern, property, or trend that has not been formally proven. It is often based on partial information, observations, and logical deduction.

The importance of conjectures lies in their ability to direct the study and exploration of mathematical theories, and they remain as conjectures until they are either proven to be true or false. In the exercise, the step of conjecturing the limit of the sequence as n approaches infinity is an example of forming a mathematical conjecture.

Partial Sums

Partial sums are used to analyze the behavior of infinite series by looking at the finite sums of their first n terms. By computing successive partial sums, one can get an insight into whether a series is converging or diverging.

Partial sums are particularly useful in understanding the sequence of the sum of series, as shown in our exercise. Since $F_n$ is defined in terms of partial sums, the comparison of partial sums played a crucial role in establishing that the sequence $\{F_n\}$ is decreasing.

Limit of a Sequence

The limit of a sequence is the value that the sequence's terms get closer to as the index goes to infinity.

If the sequence has a limit, we say it converges; otherwise, it diverges. In the situation given from our exercise example, after plotting the sequence and observing its behavior, a conjecture was made that as n approaches infinity, the limit of the sequence $\{F_n\}$ is zero. Convergent sequences like these have significant implications in fields such as analysis and number theory.