$\lim_{n \rightarrow \infty}\left(\frac{(n+1)(n+2) \ldots 3n}{n^{2n}}\right)^{1/n}$ is equal to:

JEE Main · 2016 · 03 Apr 2016 · Offline · Limits

lim_{n \to \infty} (\frac{( n + 1 ) ( n + 2 ) \dots 3 n}{n ^{2 n}})^{1/ n}

is equal to:

Answer: (A)

Let

y

be the limit and take logarithms:

ln y = lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} ln (1 + \frac{r}{n})

, the Riemann sum for

\int_{0}^{2} ln (1 + x) d x

. Substituting

t = 1 + x

, this is

\int_{1}^{3} ln t d t = [t ln t - t]_{1}^{3} = (3 ln 3 - 3) - (0 - 1) = 3 ln 3 - 2

. Since

3 ln 3 - 2 = ln \frac{27}{e ^{2}}

, the limit is

y = \frac{27}{e ^{2}}

JEE Main · 2016 · 03 Apr 2016 · Offline · Single correct option

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