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Chapter 10: Problem 67

Area Consider the sequence \(\\{1 / n\\}_{n=1}^{\infty}\) . On each subinterval \((1 /(n+1), 1 / n)\) within the interval \([0,1],\) erect the rectangle with area \(a_{n}\) having height 1\(/ n\) and width equal to the length of the subinterval. Find the total area \(\sum a_{n}\) of all the rectangles. (Hint: Use the result of Example 5 in Section \(10.2 . )\)

Short Answer

Expert verified
The total area of all rectangles is 1.

Step by step solution

01

- Understand Sequence and Intervals

We are given a sequence \(\{1/n\}_{n=1}^{\infty}\) and subintervals \((1/(n+1), 1/n)\). These subintervals are within the interval \([0,1]\). For each \(n\), the subinterval goes from \(1/(n+1)\) to \(1/n\).
02

- Determine the Width of Each Subinterval

The width of each subinterval \((1/(n+1), 1/n)\) is calculated as the difference between the endpoints: \(\frac{1}{n} - \frac{1}{n+1} = \frac{1}{n(n+1)}\).
03

- Calculate the Area of a Single Rectangle

Each rectangle has a height of \(1/n\) and a width of \(1/(n(n+1))\), so the area \(a_n\) of each rectangle is \(\text{height} \times \text{width} = \frac{1}{n} \cdot \frac{1}{n(n+1)} = \frac{1}{n^2(n+1)}\).
04

- Set Up the Series for the Total Area

We need to sum the areas of all rectangles. The total area \(\sum a_n = \sum_{n=1}^{\infty} \frac{1}{n^2(n+1)}\).
05

- Simplify the Series using Partial Fraction Decomposition

Apply partial fraction decomposition to \(\frac{1}{n^2(n+1)}\), writing it as \(\frac{A}{n} + \frac{B}{n^2} + \frac{C}{n+1}\). Solving these, we find \(A = 1, B = -1,\) and \(C = 0\), meaning \(\frac{1}{n^2(n+1)} = \frac{1}{n} - \frac{1}{n+1} - \frac{1}{n^2}\).
06

- Evaluate the Infinite Series

The series \(\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right)\) is telescopic, canceling out all but the first term, leading to \(\frac{1}{1}\), and \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) converges to a finite number (approximating to about 1.6). However, as we compare them, this means that the total area is specifically determined by the telescoping nature.
07

- Final Calculation and Conclusion

The series simplifies to \(1 - 1 + 0\), so the total area is not directly dependent on the additional series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) in terms of telescoping behavior, thus leading it to converge. By carefully considering the behavior and comparative telescoping, it is determined that the total area \(\sum a_n \) converges to 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinite Series
Infinite series are sequences of numbers that extend indefinitely. Unlike a sequence where you examine individual terms, a series involves summing all of them up. In the case of the exercise, we deal with the sum of the areas of rectangles formed under a curve. This is represented as \( \sum a_{n} \), where each \( a_{n} \) is a term from our given series. Understanding infinite series is crucial for problems involving areas and volumes because they often represent sums of infinitesimal parts. This concept is foundational in calculus and helps explain how infinite processes can result in finite quantities.
  • Series are vital in calculus for calculating areas and volumes.
  • They help express the sum of infinitely many terms.
  • Infinite series can converge to a finite value, as seen in the exercise.
Partial Fraction Decomposition
Partial fraction decomposition is a method used to break down complex rational expressions into simpler fractions. This technique is very useful when calculating sums of series like \( \sum \frac{1}{n^2(n+1)} \). By expressing a complicated fraction as the sum of easier fractions, mathematicians can more easily find and manipulate the values. In the exercise, this method is used to decompose \( \frac{1}{n^2(n+1)} \) into \( \frac{1}{n} - \frac{1}{n+1} - \frac{1}{n^2} \).
  • Decomposition makes it easier to work with algebraic expressions.
  • The technique simplifies integration and summation.
  • It turns complex fractions into a combination of simpler, manageable terms.
Telescoping Series
A telescoping series is a series where many terms cancel out with each other. This cancellation simplifies the sum to just a few terms. In the provided problem, the series formed by the decomposed fractions is telescopic, resulting in most terms cancelling each other out. The series \( \sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right) \) reduces to just a starting term of 1 and drops all others, capturing the core value.
  • Telescoping series tremendously simplify calculations.
  • The cancellation occurs across series terms, leaving only a few to sum up.
  • These series are helpful in finding finite results from infinite series.
Subintervals
Subintervals are smaller divisions of a main interval. They are significant when calculating areas under curves, especially when using rectangles to approximate these areas. In the exercise, each subinterval is given by \((1/(n+1), 1/n)\), and determines the width of the rectangles erected upon them. By understanding subintervals, you can accurately partition the main interval to better compute the total area.
  • Subintervals define where to measure each width of rectangles in approximations.
  • They are critical in approximating curves with straight-line segments like rectangles.
  • Calculations using subintervals can lead to the solution of global area estimations.
Convergence of Series
Convergence is an important concept when dealing with infinite series. A series is said to converge if the sum of its infinite terms approaches a finite limit. In our exercise, the total area is determined by this convergence, particularly through the telescoping series process. It allows us to conclude that despite the infinite number of terms, the area under the curve is finite, specifically converging to a sum of 1 in this case.
  • A series converges to a finite number if the sum of its terms levels off.
  • It is essential to determine the behavior of a series over infinite terms.
  • This concept ensures that results from infinite series make practical sense in real-world problems.

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Most popular questions from this chapter

Use the following steps to prove that the binomial series in Equation \((1)\) converges to \((1+x)^{m}\). \begin{equation}\begin{array}{l} \\ {\text { a. Differentiate the series }}\end{array}\end{equation} \begin{equation}\quad \quad \quad f(x)=1+\sum_{k=1}^{\infty} \left( \begin{array}{l}{m} \\ {k}\end{array}\right) x^{k}\end{equation} to show that \begin{equation}f^{\prime}(x)=\frac{m f(x)}{1+x}, \quad-1< x < 1.\end{equation} \begin{equation} \begin{array}{l}{\text { b. Define } g(x)=(1+x)^{-m} f(x) \text { and show that } g^{\prime}(x)=0} \\ {\text { c. From part (b), show that }}\end{array}\end{equation} \begin{equation}f(x)=(1+x)^{m}\end{equation}

Estimating Pi The English mathematician Wallis discovered the formula \begin{equation}\frac{\pi}{4}=\frac{2 \cdot 4 \cdot 4 \cdot 6 \cdot 6 \cdot 8 \cdot \cdots}{3 \cdot 3 \cdot 5 \cdot 5 \cdot 7 \cdot 7 \cdot \cdots}\end{equation} Find \(\pi\) to two decimal places with this formula.

\begin{equation} \begin{array}{l}{\text { Using a CAS, perform the following steps to aid in answering }} \\ {\text { questions (a) and (b) for the functions and intervals in Exercises }} \\ {57-62 .}\\\\{\text { Step } I : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\ {x=0}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } \int^{(n+1)}(c) \text { associat- }} \\ {\text { ed with the remainder term for each Taylor polynomial. Plot }} \\ {\text { the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approxima- }} \\ {\text { tions together. Discuss the graphs in relation to the informa- }} \\ {\text { tion discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=\frac{x}{x^{2}+1}, \quad|x| \leq 2$$

Use the definition of convergence to prove the given limit. $$ \lim _{n \rightarrow \infty}\left(1-\frac{1}{n^{2}}\right)=1 $$

\(\text { a.}\) Suppose that \(f(x)\) is differentiable for all \(x\) in \([0,1]\) and that \(f(0)=0 .\) Define sequence \(\left\\{a_{n}\right\\}\) by the rule \(a_{n}=n f(1 / n)\) Show that \(\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0) .\) Use the result in part (a) to find the limits of the following sequences \(\left\\{a_{n}\right\\} .\) $$ \begin{array}{l}{\text { b. } a_{n}=n \tan ^{-1} \frac{1}{n} \quad \text { c. } a_{n}=n\left(e^{1 / n}-1\right)} \\ {\text { d. } a_{n}=n \ln \left(1+\frac{2}{n}\right)}\end{array} $$
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