If-f-n-1-n-n-1-n-2-n-3-n-n-1-n-then-lim-n-f-n-

Question Number 22871 by keshavnimgade85@gmail.com last updated on 23/Oct/17

If f (n) = \frac{1}{n} {[(n + 1) (n + 2) (n + 3) \dots (n + n)]}^{1 / n},

then \lim_{n \to \infty} f (n) =

Answered by ajfour last updated on 23/Oct/17

f (n) = {[(1 + \frac{1}{n}) (1 + \frac{2}{n}) \dots (1 + \frac{n}{n})]}^{1 / n}

\lim_{n \to \infty} (\ln [f (n)]) = \int_{0}^{1} \ln (1 + x) d x

= x \ln (1 + x) ∣_{0}^{1} - \int_{0}^{1} \frac{x}{1 + x} d x

= \ln 2 - [x - \ln (1 + x)] ∣_{0}^{1}

= \ln 2 - 1 + \ln 2 = 2 \ln 2 - 1

\Rightarrow \lim_{n \to \infty} f (n) = e^{\ln (4 / e)} = \frac{4}{e} .

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