# P in NN\{0,1};q>0;how to solve limit?lim_(n->oo)(qn+1)/(qn)*(qn+p+1)/(qn+p)*...*(qn+np+1)/(qn+np)

Mar 27, 2017

See below.

#### Explanation:

$\frac{q n + 1}{q n} \cdot \frac{q n + p + 1}{q n + p} \cdots \frac{q n + n p + 1}{q n + n p} =$

$\left(1 + \frac{1}{n q}\right) \left(1 + \frac{1}{n q + p}\right) \cdots \left(1 + \frac{1}{n q + n p}\right) = {\prod}_{m = 0}^{n} \left(1 + \frac{1}{n q + p m}\right)$

Calling ${L}_{0} = {\lim}_{n \to \infty} {\prod}_{m = 0}^{n} \left(1 + \frac{1}{n q + p m}\right)$

applying $\log$ to both sides

$\log \left({L}_{0}\right) = {\lim}_{n \to \infty} {\sum}_{m = 0}^{n} \log \left(1 + \frac{1}{n q + m p}\right) =$

${\lim}_{n \to \infty} {\sum}_{m = 0}^{n} \log \left[{\left(1 + \frac{1}{n \left(q + \frac{m}{n} p\right)}\right)}^{n}\right] \frac{1}{n}$ and this is the Riemann-Stiltjes integral

${\lim}_{n \to \infty} {\int}_{\xi = 0}^{\xi = 1} \log \left[{\left(1 + \frac{1}{n \left(q + \xi p\right)}\right)}^{n}\right] d \xi$

so

$\log \left({L}_{0}\right) = {\lim}_{n \to \infty} \log \left({\left(\frac{1}{\frac{1}{n q} + 1}\right)}^{\frac{n q}{p}} {\left(1 + \frac{1}{n \left(p + q\right)}\right)}^{\frac{n \left(p + q\right)}{p}} {\left(\frac{1 + n \left(p + q\right)}{1 + n q}\right)}^{\frac{1}{p}}\right)$

or

$\log \left({L}_{0}\right) = \log \left({e}^{- \frac{1}{p}} {e}^{\frac{1}{p}} {\left(\frac{p + q}{q}\right)}^{\frac{1}{p}}\right) = \log \left({\left(\frac{p + q}{q}\right)}^{\frac{1}{p}}\right)$

so finally

${L}_{0} = {\left(\frac{p + q}{q}\right)}^{\frac{1}{p}}$