Question

Prove that this converges to 0:`(prod_(k=1)^n(lambdak+a)/(lambdak+b)),0<=a<b,lambda>0,n=1,2,3......`?

Answer 1

See below.

Explanation:

The product `prod_(k=1)^oo(1+a_k)` converges if and only if

the series `sum_(k=1)^oo a_k` converges and to assure `sum_(k=1)^oo a_k` convergence it is necessary that `lim_(k->oo)a_k = 0`

Now we have

`(lambdak+a)/(lambdak+b)=1+(lambdak+a-b)/(lambda k + b) = 1+ a_k`

but

`lim_(k->oo)a_k = 1` then `lim_(n->oo)(prod_(k=1)^n(lambdak+a)/(lambdak+b))` does not converges and as `0 < (lambdak+a)/(lambdak+b) < 1` we have

`lim_(n->oo)(prod_(k=1)^n(lambdak+a)/(lambdak+b)) = 0`

NOTE:

If an infinite product has a finite nonzero value, it is said to converge. Otherwise, the infinite product is said to diverge