(a) Compare appropriate areas in the accompanying figureto deduce the following
Chapter 9, Problem 32(choose chapter or problem)
(a) Compare appropriate areas in the accompanying figure to deduce the following inequalities for \(n \geq 2\) :
\(\int_{1}^{n} \ln x d x<\ln n !<\int_{1}^{n+1} \ln x d x\)
(b) Use the result in part (a) to show that
\(\frac{n^{n}}{e^{n-1}}<n !<\frac{(n+1)^{n+1}}{e^{n}}, n>1\)
(c) Use the Squeezing Theorem for Sequences (Theorem 9.1.5) and the result in part (b) to show that
\(\lim _{n \rightarrow+\infty} \frac{\sqrt[n]{n !}}{n}=\frac{1}{e}\)
Equation Transcription:
Text Transcription:
n >= 2
integral_1^n ln xdx<ln n!<integral_1^n+1 ln xdx
n^n/e^n-1<n!<(n+1)^n+1/e^n, n>1
lim n->+infinity n square root n!/n=1/e
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