Chapter 1: Q8P (page 36)

Estimate the Error if $\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}}$ is approximated by the sum of its first three terms for $| x | < \frac{1}{2}$ .

Short Answer

Expert verified

The Error is 0.001953.

Step by step solution

Given Information

The function is $\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}}$ .

Definition of the Error of the Series

The calculated Series is Underestimate if the Error is less than zero. The calculated Series is Overestimate if the Error is more than zero.

Verify the statement

$The function is :$

$\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}}$

The formula states that:

$\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}} = x + \frac{x^{2}}{2^{3}} + \frac{x^{3}}{2^{3}} + \frac{x^{4}}{2^{3}} + . . . .$

The theorem states that:

$If S = \sum_{n = 0}^{\infty} a_{n} x^{n} coverages for |x| < 1, and if |a_{n + 1}| < |a_{n}| for n > N, then |S - \sum_{n = 0}^{N} a_{n} x^{n}| < \frac{|a_{N + 1} x^{N + 1}|}{(1 - |x|)} |S - \sum_{n = 0}^{N} a_{n} x^{n}| < \frac{|a_{N + 1} x^{N + 1}|}{(1 - |x|)} |\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}} - (x + \frac{x^{2}}{2^{3}} + \frac{x^{3}}{3^{3}})| < \frac{|a_{N + 1} x^{N + 1}|}{(1 - |x|)} |\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}} - (x + \frac{x^{2}}{2^{3}} + \frac{x^{3}}{3^{3}})| < \frac{|\frac{1}{4^{3}} . \frac{1}{2^{4}}|}{1 - |\frac{1}{2}|} |\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}} - (x + \frac{x^{2}}{2^{3}} + \frac{x^{3}}{3^{3}})| < \frac{0.000976}{0.5} Solve further, and we get, |\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}} - (x + \frac{x^{2}}{2^{3}} + \frac{x^{3}}{3^{3}})| < 0.001953 Error \approx 0.001953 Hence, the Error is 0.001953 .$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Estimate the Error if $\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}}$ is approximated by the sum of its first three terms for $| x | < \frac{1}{2}$ .

Short Answer

Step by step solution

Given Information

Definition of the Error of the Series

Verify the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Solid State Physics

Astrophysics

Translational Dynamics

Electricity and Magnetism

Famous Physicists

Measurements

Study anywhere. Anytime. Across all devices.

Estimate the Error if∑n=1∞xnn3is approximated by the sum of its first three terms for|x|<12.

Short Answer

Step by step solution

Given Information

Definition of the Error of the Series

Verify the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Solid State Physics

Astrophysics

Translational Dynamics

Electricity and Magnetism

Famous Physicists

Measurements

Study anywhere. Anytime. Across all devices.

Estimate the Error if $\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{3}}$ is approximated by the sum of its first three terms for $| x | < \frac{1}{2}$ .