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 $$ \prod_{k=1}^n {(a+k)}$$ ?