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Question Number 159670 by Ar Brandon last updated on 19/Nov/21

	Answered by puissant last updated on 20/Nov/21

	$1)$ $U_{n} = \frac{n}{n^{3} + 1} \underset{+ \infty}{\sim} \frac{n}{n^{3}} = \frac{1}{n^{2}} a i n s i, s i o n c o m p a r e$ $l a s e r i e d e \frac{1}{n^{2}} a u n e s e r i e d e r i e m a n n q u i e s t$ $c o n v e r g e n t e, o n r e m a r q u e q u e c e t t e s e r i e$ $c o n v e r g e e g a l e m e n t . .$ $2)$ $U_{n} = \frac{\sqrt{n}}{n^{2} + \sqrt{n}} \underset{+ \infty}{\sim} \frac{\sqrt{n}}{n^{2}} \sim \frac{1}{n^{\frac{3}{2}}} a i n s i c e t t e$ $s e r i e e s t c o n v e r g e n t e c a r e l l e e s t$ $c o m p a r a b l e a u n e s e r i e d e R i e m a n n$ $q u i c o n v e r g e . . .$ $3)$ $U_{n} = n s i n (\frac{1}{n}) e n c a l c u l a n t l a l i m i t e,$ $o n a :$ $\lim_{n \to + \infty} n s i n (\frac{1}{n}) = \lim_{n \to + \infty} \frac{s i n (\frac{1}{n})}{\frac{1}{n}} = 1$ $e t l a s e r i e l a s e r i e d i v e r g e g r o s s i \overset{`}{e r e m e n t} . .$ $4)$ $U_{n} = \frac{1}{\sqrt{n}} l n (1 + \frac{1}{\sqrt{n}})$ $E n e f f e t, o n a : l n (1 + \frac{1}{\sqrt{n}}) \underset{+ \infty}{\sim} \frac{1}{\sqrt{n}}$ $D o n c U_{n} \underset{+ \infty}{\sim} \frac{1}{n} q u i e s t d o n c d i v e r g e n t e$ $p a r c o m p a r a i s o n a u n e s e r i e d e r i e m a n n$ $d i v e r g e n t e . .$ $5)$ $U_{n} = \frac{{(- 1)}^{n} + n}{n^{2} + 1} E n e f f e t, {(- 1)}^{n} + n \underset{+ \infty}{\sim} n$ $d e p l u s, n^{2} + 1 \underset{+ \infty}{\sim} n^{2} a i n s i, U_{n} \underset{+ \infty}{\sim} \frac{1}{n}$ $e t d i v e r g e p a r c o m p a r a i s o n a u n e s e r i e$ $d e r i e m a n n d i v e r v e n t e . . .$ $6)$ $U_{n} = \frac{1}{n!} e n f a i t, \forall n ⩾ 2, \frac{1}{n!} ⩽ \frac{1}{2^{n - 1}} a i n s i$ $p a r c o m p a r a i s o n a u n e s e r i e g e o m e t r i q u e$ $c o n v e r g e n t e, l a s e r i e c o n v e r g e . .$ $7)$ $U_{n} = \frac{3^{n} + n^{4}}{5^{n} - 2^{n}} o n r e m a r q u e a i s e m e n t q u e$ $3^{n} + n^{4} \underset{+ \infty}{\sim} 3^{n} e t 5^{n} - 2^{n} \underset{+ \infty}{\sim} 5^{n} d o n c U_{n} \underset{+ \infty}{\sim} {(\frac{3}{5})}^{n}$ $a i n s i, p a r c o m p a r a i s o n a u n e s e r i e$ $g e o m e t r i q u e c o n v e r g e n t e, l a s e r i e$ $c o n v e r g e . . .$ $. . . . . . . . . . . . . . . L e p u i s s a n t . . . . . . . . . . . . . .$
	Commented by Ar Brandon last updated on 20/Nov/21

	$Hum! merci bro .$
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