Wednesday, 13 December 2017

summation - Limit of triple sum

Suppose one has the following triple sum:



Sn=ns=0st=0su=0f(n,t)g(n,u)




where for all n, α<Sn<α for some real constant α<. Since Sn is bounded above and below by a constant may one interchange the limit with the first summand, obtaining
limnSn=s=0limn(st=0su=0f(n,t)g(n,u))?



Since the limit is now inside the first summand, may one now consider s as a constant and thus bring the limit inside the two other summands to the right of it, yielding



limnSn=s=0st=0su=0(limnf(n,t)g(n,u))?



If not, why not?

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