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

1 Answer
Feb 11, 2018

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