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 ( 10 n ) ( 2 n ) ! 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:
a n + 1 a n = 10 n + 1 ( 2 n + 2 ) ! 10 n ( 2 n ) ! = 5 ( n + 1 ) ( 2 n + 1 ) < 1
for all n 1
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Ignacio Riggs
Answered 2022-10-15 Author has 4 answers
Let a n = 10 n ( 2 n ) ! defined for natural numbers. Now, let examine the following ratio
a n + 1 a n = 10 n + 1 ( 2 ( n + 1 ) ) ! 10 n ( 2 n ) ! = 10 ( 2 n + 1 ) ( 2 n + 2 )
Now, observe that a n + 1 a n < 1 for every n 1. Thus, a n + 1 < a n which means that { a n } is decreasing sequence.
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