solve.... lim_( n→∞) { Π_(k=1) ^n (1 −(k/n)+(k^( 2) /n^( 2) ) )^( (1/n)) }=? m.n...


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Question Number 152601 by mnjuly1970 last updated on 30/Aug/21
$solve.... lim_( n→∞) { Π_(k=1) ^n (1 −(k/n)+(k^( 2) /n^( 2) ) )^( (1/n)) }=? m.n...$
$s o l v e . . . .$ ${l i m}_{n \to \infty} {\underset{k = 1}{\prod^{n}} {(1 - \frac{k}{n} + \frac{k^{2}}{n^{2}})}^{\frac{1}{n}}} = ?$ $m . n . . .$
Commented by mindispower last updated on 30/Aug/21

$h e l l o s i r i {c a n}^{'} t f i n d o n e p o s t o f y o u$ $\int \frac{s i n (x)}{x^{x}} d x s o m t h i g l i k t h a t$ $h a v e a g o o d d a y g o d b l e s s y o u$
	Answered by puissant last updated on 30/Aug/21
	$U_n =Π_(k=1) ^n (1−(k/n)+(k^2 /n^2 ))^(1/n) V_n =ln(U_n ) ⇒ lim_(n→∞) V_n = lim_(n→∞) (1/n)Σ_(k=1) ^n ln(1−(k/n)+(k^2 /n^2 )) =∫_0 ^1 ln(1−x+x^2 )dx { ((u=ln(1−x+x^2 ))),((v′=1)) :} ⇒ { ((u′=((2x−1)/(1−x+x^2 )))),((v=x)) :} =[xln(1−x+x^2 )]_0 ^1 −∫_0 ^1 ((2x^2 −x)/(1−x+x^2 ))dx To be continued...$
	$U_{n} = \underset{k = 1}{\prod^{n}} {(1 - \frac{k}{n} + \frac{k^{2}}{n^{2}})}^{\frac{1}{n}}$ $V_{n} = l n (U_{n})$ $\Rightarrow {l i m}_{n \to \infty} V_{n} = {l i m}_{n \to \infty} \frac{1}{n} \underset{k = 1}{\sum^{n}} l n (1 - \frac{k}{n} + \frac{k^{2}}{n^{2}})$ $= \int_{0}^{1} l n (1 - x + x^{2}) d x$ ${\begin{cases} u = l n (1 - x + x^{2}) \\ v^{'} = 1 \end{cases} \Rightarrow {\begin{cases} u^{'} = \frac{2 x - 1}{1 - x + x^{2}} \\ v = x \end{cases}$ $= {[x l n (1 - x + x^{2})]}_{0}^{1} - \int_{0}^{1} \frac{2 x^{2} - x}{1 - x + x^{2}} d x$ $T o b e c o n t i n u e d . . .$
	Commented by SANOGO last updated on 30/Aug/21

	$m e r c i b i e n m o n p r o f$
	Commented by puissant last updated on 30/Aug/21

	$j e s u i s p a s p r o f b r o o j e s u i s \overset{´}{e l} \overset{`}{e v e} . .$
	Commented by Ar Brandon last updated on 30/Aug/21

	$* etudiant$ 😜
	Answered by Olaf_Thorendsen last updated on 30/Aug/21

	$\int_{0}^{1} \ln (1 - x (1 - x)) d x = \lim_{n \to \infty} \frac{1}{n} \underset{k = 0}{\sum^{n}} \ln (1 - \frac{k}{n} + \frac{k^{2}}{n^{2}})$ $\int_{0}^{1} \ln (1 - x (1 - x)) d x =$ $[x \ln (1 - x + x^{2}) - 2 x - \frac{1}{2} \ln (1 - x + x^{2})$ ${+ \sqrt{3} \arctan (\frac{2 x - 1}{\sqrt{3}})]}_{0}^{1}$ $= 2 \sqrt{3} \arctan (\frac{1}{\sqrt{3}}) - 2$ $\lim_{n \to \infty} \underset{k = 1}{\prod^{n}} \ln {(1 - \frac{k}{n} + \frac{k^{2}}{n^{2}})}^{\frac{1}{n}} = e^{2 \sqrt{3} \arctan (\frac{1}{\sqrt{3}}) - 2}$
	Commented by SANOGO last updated on 30/Aug/21

	$m e r c i b i e n m o n p r o f$
	Commented by mnjuly1970 last updated on 30/Aug/21

	$t h a n k s a l o t m r o l a f . . .$
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