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

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

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

• It's an infinite product, not a series. – Robert Israel Jul 19 '13 at 4:49
• The correct terminology is : "it diverges to 0", consider taking the logarithm of this product and looking at the sum. – Arjang Jul 19 '13 at 5:26
• Thank you for correcting my amateurish mistakes. – KeithSmith Jul 19 '13 at 12:10

Consider $$P_m=\prod_{n=1}^m\frac{2^n}{3^n}=\left(\frac{2}{3}\right)^{m(m+1)/2}$$
Then, $$P_{\infty}=\prod_{n=1}^{\infty}\frac{2^n}{3^n}=\lim_{m\to\infty}P_m=\lim_{m\to\infty}\left(\frac{2}{3}\right)^{m(m+1)/2}=0$$
• You can scale rendered math for yourself if you find it too small. There's no need to use commands like \large, really. – Antonio Vargas Jul 19 '13 at 7:00