Question

`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)`

Answer 1

See below.

Explanation:

`(qn+1)/(qn)*(qn+p+1)/(qn+p)cdots(qn+np+1)/(qn+np)=`

`(1+1/(n q))(1+1/(nq+p))cdots(1+1/(nq+np)) =prod_(m=0)^n(1+1/(n q+p m))`

Calling `L_0 = lim_(n->oo)prod_(m=0)^n(1+1/(n q+p m))`

applying `log` to both sides

`log(L_0)=lim_(n->oo)sum_(m=0)^n log(1+1/(n q+m p)) =`

`lim_(n->oo)sum_(m=0)^nlog[(1+1/(n (q+m/n p)))^n]1/n` and this is the Riemann-Stiltjes integral

`lim_(n->oo)int_(xi=0)^(xi=1) log[(1+1/(n (q+xi p)))^n]d xi`

so

`log(L_0)=lim_(n->oo)log((1/(1/(nq) + 1))^((nq)/p)(1 + 1/(n(p+q)))^((n(p+q))/p)((1 + n (p + q))/(1 + n q))^(1/p))`

or

`log(L_0)=log(e^(-1/p) e^(1/p) ((p + q)/q)^(1/p)) = log(((p + q)/q)^(1/p))`

so finally

`L_0=((p + q)/q)^(1/p)`