Question #1393

I have got sequence (called An). The limit of (An) is alpha (n goes from 1 to infinite)
I have got another sequence called Xn. It looks like:
Xn=1/n*(Sum(Ak)) - (k goes from 1 to n)
I have to prove that the limit of the X sequence is alpha too!
I have tried to separate to two sequence from each other as:
lim(an*bn)=lim(an)*lim(bn)
So I think lim(1/n*(Sum(Ak))=lim(1/n)* lim(Sum(Ak))=0*lim(Sum(Ak))=0
But that's not correct because the limit of Xn should be alpha. What can I do?

Expert's answer

You have to prove that

lim_{k->}_{∞} (Sum_{k=1..n}(An)) = nα

Thus

Xn = lim(1/n*(Sum(Ak)) = 1/n* nα = α

lim

Thus

Xn = lim(1/n*(Sum(Ak)) = 1/n* nα = α

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