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The inequality
holds true for all integers
.
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Bonus Knowledge
The expression
is the product of the first
odd numbers. This can be represented as
. This fascinating mathematical property reveals how the product of odds can be directly linked to factorials, showcasing the beauty of combinatorics in a seemingly simple product!
In the context of solving the inequality
, it’s important to recognize that both sides grow quite differently as
increases. A common mistake here is to ignore the rapid growth of factorials compared to exponential functions. To effectively approach this inequality, consider analyzing the behavior at specific integer values of
or even employing Stirling’s approximation for large
to understand the comparative growth rates better!