Equivalence of the product $\prod_{i=1}^n\frac{1}{1+\frac{a}{i}}$

Question

I want to find an equialent formula of the product $$\prod_{i=1}^n\frac{1}{1+\frac{a}{i}}$$ for $ n\rightarrow\infty$.

My attempt: Using the equiavlence $$\frac{1}{1+\frac{a}{i}}\sim1-\frac{a}{i}$$ for $i\rightarrow\infty$. Now, how can I estimate the latter term? Can I say just that it is equivalent to $1$?. thank you.

Answer 1

Assuming integer $a,i,n$, $$\frac1{1+\frac ai}=\frac i{i+a}$$$$\prod\limits_{i=1}^n\frac1{1+\frac ai}=\prod\limits_{i=1}^n\frac i{i+a}={n+a\choose n}^{-1}$$

So, for general $a$, you should expect the Gamma function.