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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
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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**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 \(\epsilon\).
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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