How to calculate the this limit: $\lim _{ n\rightarrow \infty }{ \prod _{ i=1 }^{ n }{ \frac { 2i-1 }{ 2i } } } $?

Question

$\lim _{ n\rightarrow \infty }{ \prod _{ i=1 }^{ n }{ \frac { 2i-1 }{ 2i } } } $

Encountered this on the math textbook, couldn't solve it all day

Answer 1

Denote $P_n=\prod_{i=1}^n \frac{2i-1}{2i}$ your product.

Denote $Q_n=\prod_{i=1}^n \frac{2i}{2i+1}$. Note that clearly $Q_n \ge P_n \ge 0$ and $P_n \cdot Q_n=\frac{1}{2n+1}$.

Thus, $0 \le P_n \le \frac{1}{\sqrt{2n+1}}$ holds for all $n$. Now, it should be clear what the limit is...