I have been working on some homework for calc 3 and my prof has put a couple sequences in which we must find if they are increasing or decreasing with factorials in them. I've googled and there are very few examples with factorials and my prof hasn't done any examples with factorials either. I've written out what I've been able to do myself however I am not sure if it is valid, I'd appreciate some feedback.
Question:
Is the sequence {$a_r$} with $a_r=\frac{(n+1)!}{2^{n+1}}$ increasing or decreasing?
Attempt at the answer:
Test: If $a_{n+1}-a_n\ge0$ for $n\ge x$, $a_n$ is increasing
$$\frac{(n+2)!}{2^{n+2}}-\frac{(n+1)!}{2^{n+1}}\ge0$$
$$\frac{(n+2)!}{2^{n+2}}-\frac{2(n+1)!}{2^{n+2}}\ge0$$
$$\frac{(n+2)!-2(n+1)!}{2^{n+2}}\ge0$$
Here from my understanding $2^{n+2}\ge 0$ so the only way this could be negative is if $2(n+1)! > (n+2)!$ so I do the test if $\frac{(n+2)!}{2(n+1)!}\ge1$ then $a_n$ is increasing
$$\frac{1}{2}\frac{(n+2)!}{(n+1)!}\ge1$$
$$\frac{(n+2)(n+1)(n)(n-1)…3*2*1}{(n+1)(n)(n-1)…3*2*1}\ge2$$
Cancelling yields
$$n+2\ge2$$
$$n\ge0$$
Honestly I am unsure if this is a good proof, I have been taking calculus courses for my degree however I am not required to take courses that focus on writing proofs, if I could get some feedback it would be very much appreciated. There are a couple other similar questions on the assignment similar to this one, I would like to ensure I am doing this correctly before proceeding with them.
Best Answer
It's fine but instead of subtracting the terms you could try dividing them and find if $a_{n+1}/a_n\ge 1$, that would be simpler.