Prove that $\prod\limits_{i=1}^n \frac{2i-1}{2i} \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \Bbb Z_+$

Question

Given that $x_n = \displaystyle \prod_{i=1}^n \frac{2i-1}{2i}$

Then prove that $x_n \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \mathbb Z_+$

What I did was take the logarithm of $x_n$, and I arrived at: $\log{x_n}=\displaystyle \sum_{i=1}^{n} (\log{(2i-1)} - \log{2i}) $

I'd like to know if I proceeded correctly, and thus would like further guidance to solve the problem. However, if I haven't approached the problem correctly, I'd appreciate hints and techniques that are applicable. Please don't post the whole answer because I'd like to work this out on my own. Thanks.

Answer 1

Hint

Prove it by induction and you should show in the inductive step this inequality:

$$\frac{2n+1}{2n+2}\frac{1}{\sqrt{3n+1}}\le \frac{1}{\sqrt{3n+4}}$$ which is simple to see it by taking the square.

Added: Notice that $$(2n+2)^2(3n+1)-(2n+1)^2(3n+4)=n\ge0$$