Answer to Question #1393 in Complex Analysis for Norbert

Question

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?

Answer 1

You have to prove that
lim_{k->∞} (Sum_k=1..n(An)) = nα
Thus
Xn = lim(1/n*(Sum(Ak)) = 1/n* nα = α