/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 76 The series represents a well-kno... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 76

The series represents a well-known function. Use a computer algebra system to graph the partial sum \(S_{10}\) and identify the function from the graph. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{2 n+1}, \quad-1 \leq x \leq 1 $$

Short Answer

Expert verified
The given series represents the function \(f(x) = \arctan(x)\).

Step by step solution

01

Determine the power series

The given series is \(f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{2 n+1}, \quad-1 \leq x \leq 1\). Notice that it is a power series with an alternating sequence since each term in the expansion has \( (-1)^n \) which cause it to alternate between negative and positive.
02

Find the tenth partial sum for the series

The tenth partial sum, denoted as \(S_{10}\), of the given power series \(f(x)\) is essentially the sum of the first ten terms of the series. To derive it, simply replace \(n\) in the expression of the given series by 0 to 9 (inclusive), and add up these terms. We can use a computer algebra system to calculate this sum, or it can be done manually by simple substitution and addition.
03

Graph the partial sum

Next, plot the \(S_{10}\) using a computer algebra system. Do remember to plot it within the range \(-1 \leq x \leq 1\). The plotted graph will most likely represent a well-known function. Keep in mind, the more terms we include in our series, the more the shape of the graph will resemble the function.
04

Identify the function

Comparing the graphed shape with your knowledge of well-known functions will help you to identify the function the series represents. When you study the summation given, which is similar to the Taylor Series expansion of the arctan function, you can infer that the function is likely \(\arctan(x)\). Verifying it with the graph plotted earlier, the series indeed represents this function.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Alternating Series
An alternating series is a specific type of series where the signs of the terms alternate between positive and negative. This is often characterized by the presence of the term \((-1)^n\). In our exercise, the series \(f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2n+1}}{2n+1}\) is an excellent example.
Notice how \((-1)^n\) plays a role in alternating the sign of each term.
This alternation is crucial for certain mathematical properties, like convergence.
For an alternating series to converge:
  • The absolute value of the terms must decrease as the series progresses.
  • The limit of the terms, as n approaches infinity, should be zero.
In simpler terms, the terms must "settle down" to zero as you sum more of them. This helps establish stability even though the signs flip back and forth.
Partial Sum
The concept of a partial sum helps us understand how a series functions and how close it is to reaching its complete sum.
In our exercise, the partial sum \(S_{10}\) represents the sum of the first ten terms of the given series \(f(x)\).
Calculating a partial sum is like taking a snapshot of what the entire series looks like at a certain point. This involves some straightforward substitution and aggregation of terms from n = 0 to 9 in our case.
A partial sum provides a "preview" of how adding more terms progressively gets the series closer to the function it represents.
  • Start by substituting n from 0 and adding up each term until n = 9.
  • Use tools like computer algebra systems to ease the process and improve accuracy.
  • Plotting the partial sum helps visualize its resemblance to the complete function.
Understanding partial sums enhances our grasp on series by allowing us to see approximation progress as we add more terms.
Arctan Function
The arctan function, also known as the inverse tangent function, is a fundamental function with various applications in trigonometry and calculus. In our exercise, the given series is identified to represent the arctan function \(\arctan(x)\).
This function maps real numbers to angles, hence bridging the gap between algebra and geometry.
  • The series representation is: \(f(x)=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2n+1}}{2n+1}\).
  • This series mimics the Taylor Series expansion that approximates \(\arctan(x)\) over the interval \([-1, 1]\).
  • The more terms used, the closer the series' output approaches the true arctan curve.
The series and its graphical interpretation allow us to understand the behavior of \(\arctan(x)\) without needing to compute each angle directly. This makes it a powerful tool for analysis and visualization.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find all values of \(x\) for which the series converges. For these values of \(x\), write the sum of the series as a function of \(x\). $$ \sum_{n=0}^{\infty}(-1)^{n} x^{n} $$

Let \(f(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{(2 n+1) !}\) and \(g(x)=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n}}{(2 n) !} .\) (a) Find the intervals of convergence of \(f\) and \(g\). (b) Show that \(f^{\prime}(x)=g(x)\). (c) Show that \(g^{\prime}(x)=-f(x)\). (d) Identify the functions \(f\) and \(g\).

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. \(1170-\) ca. 1240 ) encountered the sequence now bearing his name. It is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}, \quad\) where \(\quad a_{1}=1\) and \(a_{2}=1\). (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by $$ b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1 . $$ (c) Using the definition in part (b), show that $$ b_{n}=1+\frac{1}{b_{n-1}} $$ (d) The golden ratio \(\rho\) can be defined by \(\lim _{n \rightarrow \infty} b_{n}=\rho\). Show that \(\rho=1+1 / \rho\) and solve this equation for \(\rho\).

Time The ball in Exercise 99 takes the following times for each fall. \(s_{1}=-16 t^{2}+16, \quad s_{1}=0\) if \(t=1\) \(s_{2}=-16 t^{2}+16(0.81), \quad s_{2}=0\) if \(t=0.9\) \(s_{3}=-16 t^{2}+16(0.81)^{2}, \quad s_{3}=0\) if \(t=(0.9)^{2}\) \(s_{4}=-16 t^{2}+16(0.81)^{3}, \quad s_{4}=0\) if \(t=(0.9)^{3}\): \(s_{n}=-16 t^{2}+16(0.81)^{n-1}, \quad s_{n}=0\) if \(t=(0.9)^{n-1}\) Beginning with \(s_{2}\), the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is given by \(t=1+2 \sum_{n=1}^{\infty}(0.9)^{n}\) Find this total time.

Explain how to use the series \(g(x)=e^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\) to find the series for each function. Do not find the series. (a) \(f(x)=e^{-x}\) (b) \(f(x)=e^{3 x}\) (c) \(f(x)=x e^{x}\) (d) \(f(x)=e^{2 x}+e^{-2 x}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.