Real analysis Exercises 12. 1. Compute lim (n!)¹/nn(Hint: let sn=n!in Theorem 12.3 and recall that lim (1 + 1/ )" = e)nnn Theorem 12.3. Let (sn) be a non-zero sequence. Thenlim inf ≤ lim inf|s₁|¹/n ≤ lim sup s₁¹/"

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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

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter3: The Derivative

Section3.1: Limits

Problem 63E

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Real analysis

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

Theorem 12.3. Let (sn) be a non-zero sequence. Then
lim inf ≤ lim inf|s₁|¹/n ≤ lim sup s₁¹/" <lim sup
Sn
|1/n
Sn+1
Sn
In particular, lim = L ⇒ lim |s₂|¹/n = L
Sn+1
Sn
Sn+1
Sn

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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