How to show this sequence is decreasing

Question

I was given a practice question in my math lecture today, it was to prove that the sequence below was decreasing:

$a_n = \frac{2^n}{(n+1)!}$

using the condition that a sequence is decreasing if $a_{n+1} \leq a_n$

So far I have found that

$a_{n+1} = \frac{2^{n+1}}{(n+2)!}$

but I dont know how to prove this other than plugging in numbers and just showing that $a_{n+1}$ is less than $a_n$. Any help would be much appreciated

Answer 1

Note that$$\frac{a_{n+1}}{a_n}=\frac2{n+2}<1$$and that therefore $a_{n+1}<a_n$.

Answer 2

Note that

$$\frac{a_{n+1}}{a_n}=\frac{2^{n+1}}{(n+2)!}\frac{(n+1)!}{2^n}=\frac{2}{n+2}< 1$$

or as an alternative

$$a_n-a_{n+1}=\frac{2^n}{(n+1)!}-\frac{2^{n+1}}{(n+2)!}=\frac{2^n(n+2)-2^{n+1}}{(n+2)!}=2^n\frac{n}{(n+2)!}> 0$$

Answer 3

$$a_{n+1} = \frac{2^{n+1}}{(n+2)!} = \underbrace{\frac{2}{n+2}}_{< 1}\cdot\underbrace{\frac{2^n}{(n+1)!}}_{=a_n} < a_n$$