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Exercises 12. 1. Compute lim 1 (n!)¹/n (Hint: let Sn = in Theorem 12.3 and recall that lim (1+1)" = e) nn

Exercises 12. 1. Compute lim 1 (n!)¹/n (Hint: let Sn = in Theorem 12.3 and recall that lim (1+1)" = e) nn

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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Real analysis

**Exercises 12.** 1. Compute \(\lim \frac{1}{n}(n!)^{1/n}\) *(Hint: let \(s_n = \frac{n!}{n^n}\) in Theorem 12.3 and recall that \(\lim \left(1 + \frac{1}{n}\right)^n = e\))*
Transcribed Image Text:**Exercises 12.** 1. Compute \(\lim \frac{1}{n}(n!)^{1/n}\) *(Hint: let \(s_n = \frac{n!}{n^n}\) in Theorem 12.3 and recall that \(\lim \left(1 + \frac{1}{n}\right)^n = e\))*
**Theorem 12.3.** Let \((s_n)\) be a non-zero sequence. Then \[ \liminf \left|\frac{s_{n+1}}{s_n}\right| \leq \liminf |s_n|^{1/n} \leq \limsup |s_n|^{1/n} \leq \limsup \left|\frac{s_{n+1}}{s_n}\right| \] In particular, \(\lim \left|\frac{s_{n+1}}{s_n}\right| = L \implies \lim |s_n|^{1/n} = L\)
Transcribed Image Text:**Theorem 12.3.** Let \((s_n)\) be a non-zero sequence. Then \[ \liminf \left|\frac{s_{n+1}}{s_n}\right| \leq \liminf |s_n|^{1/n} \leq \limsup |s_n|^{1/n} \leq \limsup \left|\frac{s_{n+1}}{s_n}\right| \] In particular, \(\lim \left|\frac{s_{n+1}}{s_n}\right| = L \implies \lim |s_n|^{1/n} = L\)
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