# 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.

Eliza Gregory 2022-10-13 Answered
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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## Answers (2)

Martha Dickson
Answered 2022-10-14 Author has 20 answers
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
Answered 2022-10-15 Author has 4 answers
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.
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