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$1cdot3cdot5cdots(2n-1) < (frac{2n}{e})^{n+1}, n in mathbb{N}, n geq 2$ proof doesn't seem to work

Mathematics Asked by testcase12 on January 22, 2021

I found that excercise in an old book without any hint nor solution. Olthough I know a thing ot two about induction, this one seems to be too tricky for me. I know for sure that it’s a proof by induction

It is known that: $3e^3 < 64$ (for the base case)
Prove that:

$$1cdot3cdot5cdots(2n-1) < left( frac{2n}{e} right)^{n+1}, quad n in mathbb{N}, n ge 2$$

The induction that I am trying to do looks fine in terms of calculations, yet it is not proving anything.

One Answer

Without induction, the problem would be quite simple since you want to prove that $$frac{2^n }{sqrt{pi }}Gamma left(n+frac{1}{2}right)<left( frac{2n}{e} right)^{n+1}$$ Taking logarithms and using Stirling approximation $$log(text{rhs - lhs})=left(log (n)-1+frac{log (2)}{2}right)+frac{1}{24 n}+Oleft(frac{1}{n^3}right)$$ $$text{rhs - lhs}=frac{sqrt{2} n}{e}+frac{1}{12 sqrt{2} e}+frac{1}{576 sqrt{2} e n}+Oleft(frac{1}{n^2}right)$$

Answered by Claude Leibovici on January 22, 2021

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