lim_(n→∞) ((( ((n),(0) ) ((n),(1) ) ((n),(2) )... ((n),(n) )))^(1/(n^2 +n)) )


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Question Number 42367 by Joel578 last updated on 24/Aug/18

$\lim_{n \to \infty} (\sqrt[n^{2} + n]{(\begin{matrix} n \\ 0 \end{matrix}) (\begin{matrix} n \\ 1 \end{matrix}) (\begin{matrix} n \\ 2 \end{matrix}) . . . (\begin{matrix} n \\ n \end{matrix})})$
	Answered by tanmay.chaudhury50@gmail.com last updated on 24/Aug/18
	$T_n =nc_0 ×nc_1 ×nc_2 ...×nc_n T_n =((n!)/(0!(n−0)!))×((n!)/(1!(n−1)!))×((n!)/(2!(n−2)!))×...×((n!)/(n!(n−n)!)) T_n =(((n!)^n )/((0!×1!×2!×3!..×n!){(n−0)!×(n−1)!×(n−2)!×..×(n−n)!})) T_n =(((n!)^2 )/((1!×2!×3!...×n!)^2 )) T_n =(1/((((1!)/(n!))×((2!)/(n!))×((3!)/(n!))×...×((n!)/(n!)))^2 )) lim_(n→∞) [(1/((((1!)/(n!))×((2!)/(n!))×((3!)/(n!))×...×((n!)/(n!)))^2 ))]^(1/(n^2 +n)) contd...$
	$T_{n} = {n c}_{0} \times {n c}_{1} \times {n c}_{2} . . . \times {n c}_{n}$ $T_{n} = \frac{n!}{0! (n - 0)!} \times \frac{n!}{1! (n - 1)!} \times \frac{n!}{2! (n - 2)!} \times . . . \times \frac{n!}{n! (n - n)!}$ $T_{n} = \frac{{(n!)}^{n}}{(0! \times 1! \times 2! \times 3! . . \times n!) {(n - 0)! \times (n - 1)! \times (n - 2)! \times . . \times (n - n)!}}$ $T_{n} = \frac{{(n!)}^{2}}{{(1! \times 2! \times 3! . . . \times n!)}^{2}}$ $T_{n} = \frac{1}{{(\frac{1!}{n!} \times \frac{2!}{n!} \times \frac{3!}{n!} \times . . . \times \frac{n!}{n!})}^{2}}$ $l i \underset{n \to \infty}{m} {[\frac{1}{{(\frac{1!}{n!} \times \frac{2!}{n!} \times \frac{3!}{n!} \times . . . \times \frac{n!}{n!})}^{2}}]}^{\frac{1}{n^{2} + n}}$ $c o n t d . . .$
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