# Determine whether the Sequence is decreasing or increasing. I have the sequence ((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.

Determine whether the Sequence is decreasing or increasing.
I have the sequence $\frac{\left({10}^{n}\right)}{\left(2n\right)!}$ 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.
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Martha Dickson
The hint:
$\frac{{a}_{n+1}}{{a}_{n}}=\frac{\frac{{10}^{n+1}}{\left(2n+2\right)!}}{\frac{{10}^{n}}{\left(2n\right)!}}=\frac{5}{\left(n+1\right)\left(2n+1\right)}<1$
for all $n\ge 1$
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Ignacio Riggs
Let ${a}_{n}=\frac{{10}^{n}}{\left(2n\right)!}$ defined for natural numbers. Now, let examine the following ratio
$\frac{{a}_{n+1}}{{a}_{n}}=\frac{\frac{{10}^{n+1}}{\left(2\left(n+1\right)\right)!}}{\frac{{10}^{n}}{\left(2n\right)!}}=\frac{10}{\left(2n+1\right)\left(2n+2\right)}$
Now, observe that $\frac{{a}_{n+1}}{{a}_{n}}<1$ for every $n\ge 1$. Thus, ${a}_{n+1}<{a}_{n}$ which means that $\left\{{a}_{n}\right\}$ is decreasing sequence.