- #1

evinda

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MHB

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Consider the sequence $a_n=\frac{8^n}{n!}$.

It holds that

$$a_{n+1}-a_n=\frac{8^{n+1}}{(n+1)!}-\frac{8^n}{n!}=\frac{8 \cdot 8^n}{(n+1) \cdot n!}-\frac{8^n}{n!}=\frac{8^n}{n!}\left( \frac{8}{n+1}-1\right)=\frac{8^n}{n!} \left( \frac{7-n}{n+1}\right)$$

Since the last term is positive for some $n$ and negative for others, we cannot conclude like that if the sequece is monotonic or not.

I haven't thought of an other criterion which we could use to see if the sequence is monotonic or not. Could you give me a hint? (Thinking)