I would like study the convergence of the following sequence:

$$ u\_{n}=\sqrt [n]{\frac{(a+1)(a+2)...(a+n)}{n!}} $$

where $a>0$

We have:

$$ \ln(u\_{n})=\frac{1}{n}\sum\_{k=1}^n \ln(1+\frac{a}{k})$$

And : $$ \ln(u\_{n+1})-\ln(u\_{n})=\frac{1}{n+1}\sum\_{k=1}^{n+1} \ln(1+\frac{a}{k})-\frac{1}{n}\sum\_{k=1}^n \ln(1+\frac{a}{k})$$ 

So I have to study the convergence of $$ \sum \ln(u\_{n+1})-\ln(u\_{n})$$ 

Using integrals: $$ \sum\_{k=1}^n \ln(1+\frac{a}{k}) \sim a\ln(n)$$

Thus: $$ \ln(u\_{n+1})-\ln(u\_{n})= \frac{1}{n}a\ln(n)-\frac{1}{n}a\ln(n)+o(\frac{\ln(n)}{n})=o(\frac{\ln(n)}{n}) $$

which is not enough to determine the convergence or divergence of $ \sum \ln(u\_{n+1})-\ln(u\_{n})$

So how can I find a more precise approximation of $ \ln(u\_{n+1})-\ln(u\_{n})$?

Or is there a simple equivalent of $$ \prod\_{k=1}^n {(a+k)}$$ ?