Infinite product of consecutive odd/even, even/odd ratios

[Loren Booda]

$$?=\prod_{n=1}^\infty(n/(n+1))$$

 

[CRGreathouse]

Let
$$P(N)=\prod_{n=1}^N\frac{n}{n+1}.$$

Then you're looking for $\lim_{n\to\infty}P(n).$

P(1) = 1/2, P(2) = 1/2 * 2/3 = 1/3, P(3) = ... do you see what's going on here? What's the limit going to be?

 

[rrronny]

$$\lim_{N \to \infty}P(N) = \lim_{N \to \infty} \frac{1}{N+1} = 0.$$

 

[Loren Booda]

Thank you, CR. I just needed a little shaking up.