Infinite product limit and estimate

Question

I came across this product series in research and need to understand and estimate it. It appears to be unbound, but what would be the law? $$\lim_{n \to \infty} \prod_{k=1}^{n}\frac1{(1 - \frac1{2k+1})}$$ Many thanks if you know the answer.

Answer 1

The first thing I would do is write the product term to accurately reflect the fact that multiplication is taking place over odds: $$\lim_{n \to \infty} \prod_{k=1}^{n}\frac{2k+1}{2k}=\lim_{n \to \infty} \prod_{k=1}^{n}\left(1+\frac{1}{2k}\right)$$ Regarding convergence, a useful theorem says if $a_k>0$, then $\sum a_k$ and $\prod (1+a_k)$ converge or diverge together: not to the same value, but one is finite if and only if the other is. So we can study $$\frac{1}{2} \sum _{k=1}^{\infty} \frac{1}{k}, $$a well-known series known to be divergent called the harmonic series. Your intuition is correct, i.e. the product is unbounded (does not converge).