How to find $ \lim_{n \to \infty}\prod_{i=0}^{n}\frac {qn+ip+1}{qn+ip}$?

Question

How to find

$$L=\lim_{n \to \infty}\prod_{i=0}^{n}\frac {qn+ip+1}{qn+ip}\,,$$

where $p\in\Bbb N ,p \neq \{0,1\},q>0$?

Also, I'm bound to using elemental methods.

Can you give some context to the question, in what context did it appear? — OneAndOnlyDaniel, Jun 02 '19 at 15:53
@OneAndOnlyDaniel It's not part of a bigger exercise or a specific chapter. The only context that I have is that it's part of calculus, which doesn't help that much. — Rareș Stanca, Jun 02 '19 at 15:56

score 1 · Answer 1 · answered Jun 02 '19 at 16:00

Hint: Take the $\ln$ on both sides:

$$\ln L=\lim_{n\to\infty}\sum_{i=0}^n\ln\left(\frac{qn+ip+1}{qn+ip}\right)$$

$$\implies L=\exp\left(\lim_{n\to\infty}\sum_{i=0}^n\ln\left(1+\frac{1}{qn+ip}\right)\right)$$

Now, you have reduced the problem to an infinite series.

1 Answers1