Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
1 {a,}={ 4. Let | 2n² Let ɛ =.001. Find a natural number N such that |a, - L<ɛ when n 2 N.

1 {a,}={ 4. Let | 2n² Let ɛ =.001. Find a natural number N such that |a, - L<ɛ when n 2 N.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

need assitance if possible 

**Problem Statement:** 4. Let \( \{ a_n \} = \left\{ \frac{1}{2n^2} \right\} \). Let \( \varepsilon = 0.001 \). Find a natural number \( N \) such that \( | a_n - L | < \varepsilon \) when \( n \geq N \). **Explanation:** This problem deals with sequences and limits. The sequence is defined as \( \{ a_n \} = \left\{ \frac{1}{2n^2} \right\} \). The task is to find a natural number \( N \) such that for all \( n \) greater than or equal to \( N \), the absolute difference between the sequence term \( a_n \) and the limit \( L \) is less than the small positive number \( \varepsilon = 0.001 \). The underlying concept is the definition of convergence in sequences, where \( L \) is the limit that the sequence approaches as \( n \) becomes very large. In this case, the limit \( L \) is 0, because as \( n \) approaches infinity, \( \frac{1}{2n^2} \) approaches 0. Therefore, the goal is to identify \( N \) such that for every \( n \geq N \), the inequality \( \left| \frac{1}{2n^2} - 0 \right| < 0.001 \) holds.
Transcribed Image Text:**Problem Statement:** 4. Let \( \{ a_n \} = \left\{ \frac{1}{2n^2} \right\} \). Let \( \varepsilon = 0.001 \). Find a natural number \( N \) such that \( | a_n - L | < \varepsilon \) when \( n \geq N \). **Explanation:** This problem deals with sequences and limits. The sequence is defined as \( \{ a_n \} = \left\{ \frac{1}{2n^2} \right\} \). The task is to find a natural number \( N \) such that for all \( n \) greater than or equal to \( N \), the absolute difference between the sequence term \( a_n \) and the limit \( L \) is less than the small positive number \( \varepsilon = 0.001 \). The underlying concept is the definition of convergence in sequences, where \( L \) is the limit that the sequence approaches as \( n \) becomes very large. In this case, the limit \( L \) is 0, because as \( n \) approaches infinity, \( \frac{1}{2n^2} \) approaches 0. Therefore, the goal is to identify \( N \) such that for every \( n \geq N \), the inequality \( \left| \frac{1}{2n^2} - 0 \right| < 0.001 \) holds.
Expert Solution
Step 1:Limit.

Advanced Math homework question answer, step 1, image 1

 

steps

Step by step

Solved in 2 steps with 3 images

Blurred answer
Knowledge Booster
Ratios
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS