Prove that the following sequence is 22 1+ + + + 2! 3! Sk = a Cauchy sequence, 3² k² k! " KEN.
Prove that the following sequence is 22 1+ + + + 2! 3! Sk = a Cauchy sequence, 3² k² k! " KEN.
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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![**Prove that the following sequence is a Cauchy sequence:**
\[
s_k = 1 + \frac{2^2}{2!} + \frac{3^2}{3!} + \cdots + \frac{k^2}{k!}, \quad k \in \mathbb{N}.
\]
**Explanation:**
The sequence \(s_k\) is defined as the summation of terms of the form \(\frac{n^2}{n!}\) where \(n\) ranges from 1 to \(k\). A sequence is said to be a Cauchy sequence if, for every positive number \(\epsilon\), there exists an integer \(N\) such that for all integers \(m, n > N\), the absolute difference \(|s_m - s_n|\) is less than \(\epsilon\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a04a0c8-a6a9-473e-b4ed-0840bde177a1%2Fccd6eeb9-6d97-4e11-b9f3-a4856dad253b%2Fyeqweil_processed.png&w=3840&q=75)
Transcribed Image Text:**Prove that the following sequence is a Cauchy sequence:**
\[
s_k = 1 + \frac{2^2}{2!} + \frac{3^2}{3!} + \cdots + \frac{k^2}{k!}, \quad k \in \mathbb{N}.
\]
**Explanation:**
The sequence \(s_k\) is defined as the summation of terms of the form \(\frac{n^2}{n!}\) where \(n\) ranges from 1 to \(k\). A sequence is said to be a Cauchy sequence if, for every positive number \(\epsilon\), there exists an integer \(N\) such that for all integers \(m, n > N\), the absolute difference \(|s_m - s_n|\) is less than \(\epsilon\).
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