Does the infinite product $\prod_{n \mathop = 1}^\infty {\frac{2^n}{3^n}}$ diverge to zero or some other finite value.

Question

Does the infinite product diverge to zero or some other value?

$$\prod_{n \mathop = 1}^\infty {\frac{2^n}{3^n}}$$

Answer 1

Consider $$P_m=\prod_{n=1}^m\frac{2^n}{3^n}=\left(\frac{2}{3}\right)^{m(m+1)/2}$$

Then, $$P_{\infty}=\prod_{n=1}^{\infty}\frac{2^n}{3^n}=\lim_{m\to\infty}P_m=\lim_{m\to\infty}\left(\frac{2}{3}\right)^{m(m+1)/2}=0$$