find-lim-n-n-1-1-n-1-2-n-1-n-n-

Question Number 30755 by abdo imad last updated on 25/Feb/18

f i n d {l i m}_{n \to \infty}^{n} \sqrt{(1 + \frac{1}{n}) (1 + \frac{2}{n}) \dots (1 + \frac{n}{n})}

Commented by abdo imad last updated on 28/Feb/18

l e t p u t A_{n} =^{n} \sqrt{(1 + \frac{1}{n}) (1 + \frac{2}{n}) \dots . (1 + \frac{n}{n})} w e h a v e

l n (A_{n}) = \frac{1}{n} l n (\prod_{k = 1}^{n} (1 + \frac{k}{n})) = \frac{1}{n} \sum_{k = 1}^{n} l n (1 + \frac{k}{n}) s o

l n (A_{n}) i s a R i e m a n s u m a n d {l i m}_{n \to \infty} l n (A_{n})

= \int_{0}^{1} l n (1 + x) d x = \int_{1}^{2} l n (t) d t = {[t l n t - t]}_{1}^{2}

= 2 l n 2 - 2 + 1 = 2 l n 2 - 1 \Rightarrow {l i m}_{n \to \infty} A_{n} = e^{- 1} e^{l n (4)} = \frac{4}{e} .