which of the following is increasing or decreasing a. u_n = ((n!)/n^n ) b. u_n = (4^n /(3^n +1)) c. u


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Question Number 78569 by Rio Michael last updated on 18/Jan/20

$w h i c h o f t h e f o l l o w i n g i s i n c r e a s i n g o r d e c r e a s i n g$ $a . u_{n} = \frac{n!}{n^{n}}$ $b . u_{n} = \frac{4^{n}}{3^{n} + 1}$ $c . u_{n} = \frac{2^{n}}{n^{2}}$
	Answered by mr W last updated on 18/Jan/20

	$a)$ $\frac{u_{n + 1}}{u_{n}} = \frac{(n + 1)! n^{n}}{{(n + 1)}^{n + 1} n!} = {(\frac{n}{n + 1})}^{n} < 1$ $\Rightarrow d e c r e a s i n g$ $b)$ $\frac{u_{n + 1}}{u_{n}} = \frac{4^{n + 1}}{3^{n + 1} + 1} \times \frac{3^{n} + 1}{4^{n}} = 1 + \frac{3^{n} + 3}{3 \times 3^{n} + 1} > 1$ $\Rightarrow i n c r e a s i n g$ $c)$ $\frac{u_{n + 1}}{u_{n}} = \frac{2^{n + 1}}{{(n + 1)}^{2}} \times \frac{n^{2}}{2^{n}} = 2 {(\frac{n}{n + 1})}^{2} > 1 u p o n n = 3$ $\Rightarrow i n c r e a s i n g u p o n n = 3$
	Commented by Rio Michael last updated on 19/Jan/20

	$t h a n k s s i r$
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