Question-172189

Question Number 172189 by Mikenice last updated on 23/Jun/22

Answered by Mathspace last updated on 24/Jun/22

l e t u_{n} = \prod_{k = 1}^{n} {(φ (a + \frac{k b}{n}))}^{\frac{1}{n}}

\Rightarrow {l i m}_{n \to + \infty} l n (u_{n})

= {l i m}_{n \to + \infty} \frac{1}{n} l n (φ (a + \frac{k b}{n})

= \frac{1}{b} {l i m}_{n \to + \infty} \frac{a + b - a}{n} l n (φ (a + \frac{k (a + b - a)}{n}))

= \frac{1}{b} \int_{a}^{a + b - a} l n (φ (x)) d x = λ \Rightarrow

{l i m}_{n \to + \infty} u_{n} = e^{λ}