Use the definition of the limit of a sequence to prove that lim (Vn + 2021 – vn) = 0. that jat- %3D
Use the definition of the limit of a sequence to prove that lim (Vn + 2021 – vn) = 0. that jat- %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement**
5. Use the definition of the limit of a sequence to prove that
\[
\lim_{{n \to \infty}} (\sqrt{n + 2021} - \sqrt{n}) = 0.
\]
**Explanation**
This problem asks you to prove that the limit of the sequence defined by the expression \((\sqrt{n + 2021} - \sqrt{n})\) approaches 0 as \(n\) approaches infinity, using the formal definition of a limit for sequences. This involves demonstrating that for every positive number \(\epsilon\), there exists a positive integer \(N\) such that for all \(n > N\), the absolute difference between the sequence term and the limit (0 in this case) is less than \(\epsilon\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80fac62b-1f91-4b94-88b2-6d2c841a4645%2Ffb027703-ecaa-4044-b208-04f00a1d18ab%2Fzsuopu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
5. Use the definition of the limit of a sequence to prove that
\[
\lim_{{n \to \infty}} (\sqrt{n + 2021} - \sqrt{n}) = 0.
\]
**Explanation**
This problem asks you to prove that the limit of the sequence defined by the expression \((\sqrt{n + 2021} - \sqrt{n})\) approaches 0 as \(n\) approaches infinity, using the formal definition of a limit for sequences. This involves demonstrating that for every positive number \(\epsilon\), there exists a positive integer \(N\) such that for all \(n > N\), the absolute difference between the sequence term and the limit (0 in this case) is less than \(\epsilon\).
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