Determine whether the Sequence is decreasing or increasing.

Question

I have the sequence $\frac{(10^n)}{(2n)!}$ and am trying to determine whether the sequence decreases or increases. I feel like the best way to proceed would be to use the squeeze theorem, but am unsure how to apply it to the problem.

Answer 1

Let $a_n=\frac{10^n}{(2n)!}$ defined for natural numbers. Now, let examine the following ratio

$$\frac{a_{n+1}}{a_n}=\frac{\frac{10^{n+1}}{(2(n+1))!}}{\frac{10^n}{(2n)!}}=\frac{10}{(2n+1)(2n+2)}$$

Now, observe that $\frac{a_{n+1}}{a_n}<1$ for every $n\geq 1$. Thus, $a_{n+1}<a_n$ which means that $\{a_n\}$ is decreasing sequence.

Answer 2

The hint: $$\frac{a_{n+1}}{a_n}=\frac{\frac{10^{n+1}}{(2n+2)!}}{\frac{10^n}{(2n)!}}=\frac{5}{(n+1)(2n+1)}<1$$ for all $n\geq1$.