/*! 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 49 Find the sum of the convergent s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 49

Find the sum of the convergent series by using a well-known function. Identify the function and explain how you obtained the sum. $$ \sum_{n=0}^{\infty}(-1)^{n} \frac{1}{2^{2 n+1}(2 n+1)} $$

Short Answer

Expert verified
The sum of the series is approximately 0.245, which is the result of \( Arctan(\frac{1}{4}) \).

Step by step solution

01

Identify The Series Structure

This series resembles the Taylor Series expansion of the Arctan function: \( Arctan(x) = \sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2n+1}}{2n+1} \). In our given series, x is replaced by \( \frac{1}{2^2} \) or \( \frac{1}{4} \). So, we can rewrite the series as \( \sum_{n=0}^{\infty}(-1)^{n} \frac{(1/4)^{n+1}}{2n+1} \).
02

Apply Known Function To Series

By comparing the series with the Taylor series of the Arctan function, it is known that our series is a specific representation of the Arctan function that applies when \( x = \frac{1}{4} \). Therefore, calculating the sum of the series equates to evaluating \( Arctan(\frac{1}{4}) \).
03

Calculate The Function

The result can be found in any scientific calculator or a tool such as Wolfram Alpha, this gives the result of \( Arctan(\frac{1}{4}) \approx 0.245 \).

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!

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

Inflation If the rate of inflation is \(4 \frac{1}{2} \%\) per year and the average price of a car is currently \(\$ 16,000,\) the average price after \(n\) years is \(P_{n}=\$ 16,000(1.045)^{n}\) Compute the average prices for the next 5 years.

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{100} $$

Compound Interest Consider the sequence \(\left\\{A_{n}\right\\}\) whose \(n\) th term is given by \(A_{n}=P\left(1+\frac{r}{12}\right)^{n}\) where \(P\) is the principal, \(A_{n}\) is the account balance after \(n\) months, and \(r\) is the interest rate compounded annually. (a) Is \(\left\\{A_{n}\right\\}\) a convergent sequence? Explain. (b) Find the first 10 terms of the sequence if \(P=\$ 9000\) and \(r=0.055\)

Find the sum of the convergent series. $$ \sum_{n=1}^{\infty}\left[(0.7)^{n}+(0.9)^{n}\right] $$

Conjecture Let \(x_{0}=1\) and consider the sequence \(x_{n}\) given by the formula \(x_{n}=\frac{1}{2} x_{n-1}+\frac{1}{x_{n-1}}, \quad n=1,2, \ldots .\) Use a graphing utility to compute the first 10 terms of the sequence and make a conjecture about the limit of the sequence.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

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.