lim_(x→∞) ((∫_0 ^1 (1+x^n )^n dx))^(1/n) =?


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Question Number 86675 by M±th+et£s last updated on 30/Mar/20

$\underset{x \to \infty}{l i m} \sqrt[n]{\int_{0}^{1} {(1 + x^{n})}^{n} d x} = ?$
Commented by M±th+et£s last updated on 31/Mar/20

$\underset{n \to \infty}{l i m} * * *$
Commented by mr W last updated on 30/Mar/20

$= 1$
Commented by redmiiuser last updated on 30/Mar/20

$h o w s i r ?$
Commented by mr W last updated on 30/Mar/20

$t h r o u g h t h i n k i n g . . .$
Commented by mathmax by abdo last updated on 31/Mar/20

$l e t t a k e a t r y A_{n} = {(\int_{0}^{1} {(1 + x^{n})}^{n} d x)}^{\frac{1}{n}} \Rightarrow$ $l n (A_{n}) = \frac{1}{n} \int_{0}^{1} {(1 + x^{n})}^{n} d x = \frac{1}{n} \int_{0}^{1} (\sum_{k = 0}^{n} x^{n k}) d x$ $= \frac{1}{n} \sum_{k = 0}^{n} \frac{1}{n k + 1} {[x^{n k + 1}]}_{0}^{1} d x = \frac{1}{n} \sum_{k = 0}^{n} \frac{1}{n k + 1} x d i s a p p e a r$ $s o m e t h i n g w e n t w r o n g i n t h e Q . . .!$
Commented by mr W last updated on 31/Mar/20

$s h o u l d i t n o t b e :$ $l n (A_{n}) = \frac{1}{n} \ln [\int_{0}^{1} {(1 + x^{n})}^{n} d x] i n s t e a d o f$ $l n (A_{n}) = \frac{1}{n} \int_{0}^{1} {(1 + x^{n})}^{n} d x ?$
Commented by mathmax by abdo last updated on 01/Apr/20

$y e s s i r .$
	Answered by redmiiuser last updated on 30/Mar/20

	${(1 + x^{n})}^{n}$ $= \underset{k = 0}{\sum^{n}} . \underset{k}{\overset{n}{C}} . {(x^{n})}^{k}$ $\underset{0}{\int^{1}} \underset{k = 0}{\sum^{n}} . \underset{k}{\overset{n}{C}} . x^{n k} . d x$ $= \underset{k = 0}{\sum^{n}} . \underset{k}{\overset{n}{C}} . {[\frac{x^{n k + 1}}{n k + 1}]}_{0}^{1}$ $= \underset{k = 0}{\sum^{n}} . \underset{k}{\overset{n}{C}} . [\frac{1}{n k + 1}]$ $∴ {[\underset{k = 0}{\sum^{n}} . \underset{k}{\overset{n}{C}} . [\frac{1}{n k + 1}]]}^{(1 / n)} \frac{}{} a n s w e r$
	Commented by redmiiuser last updated on 30/Mar/20

	$s i r p l s c h e c k t h e a n s$
	Commented by M±th+et£s last updated on 30/Mar/20

	$g o d b l e s s y o u s i r . i t s c o r r e c t s i r b u t h o w c a n w e f i n d$ $\underset{x \to \infty}{l i m} \sqrt{a n s}$
	Commented by redmiiuser last updated on 30/Mar/20

	$n o t e r m o f x i s p r e s e n t i n$ $m y a n s w e r$ $h e n c e t h a t l i \underset{x \to \infty}{m} a n s$ $m a k e s n o s e n s e .$
	Commented by M±th+et£s last updated on 30/Mar/20

	$s o s o r r y i m e a n \underset{n \to 0}{l i m} a n s$
	Commented by M±th+et£s last updated on 30/Mar/20

	$\underset{n \to \infty}{l i m} * * *$
	Commented by redmiiuser last updated on 30/Mar/20

	$i t s o k . h o p e y o u h a d$ $g o t t h e c o n c e p t .$
	Commented by redmiiuser last updated on 30/Mar/20

	$o k . n o w i g o t i t .$
	Commented by redmiiuser last updated on 30/Mar/20

	$a r e y o u s u r e t h a t i t i s$ $l i \underset{n \to \infty}{m} * * *$
	Commented by M±th+et£s last updated on 30/Mar/20

	$y e s s i r$
	Commented by redmiiuser last updated on 30/Mar/20

	$t h e n l e t m e t r y .$
	Commented by redmiiuser last updated on 30/Mar/20

	$l i \underset{n \to \infty}{m} \underset{k = 0}{\sum^{n}} . \underset{k}{\overset{n}{C}} . {(\frac{1}{n k + 1})}^{(1 / n)}$ $= \underset{k = 0}{\sum^{\infty}} . \underset{k}{\overset{\infty}{C}} . {(\frac{1}{\infty k + 1})}^{(1 / \infty)}$ $c l e a r l y u n d e f i n e d .$
	Commented by redmiiuser last updated on 30/Mar/20

	$a s 0^{0} i s m e a n i n g l e s s .$ $a s w e l l a s I {d o n}^{'} t t h i n k$ $t h e r e e x s i s t s \underset{k}{\overset{\infty}{C}} .$
	Commented by redmiiuser last updated on 30/Mar/20

	$s i r c a n y o u c h e c k i t ?$
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