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Limit of a sequence

  1. Sep 5, 2011 #1
    1. The problem statement, all variables and given/known data
    let (Xn) be a sequence in R. Let (An) be a sequence defined as An=(X1 +X2+...Xn-1+Xn)/n. (Xn) is a convergent sequence and the limit of Xn as n goes to infinity is L. Prove (An) in convergent sequence and that the limit is also L.

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 5, 2011 #2

    SammyS

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    What have you tried? Where are you stuck.

    Write out the first few terms of (An).
     
  4. Sep 5, 2011 #3
    A1=X1, A2= (X1+X2)/2, A3=(X1+X2+X3)/3.... I HAVE A THM IN MY BOOK THAT SAYS IF THE SUMMATION OF A SEQUENCE Xn CONVERGES THEN THE LIMIT OF Xn is 0. So I first assumed that the limit of Xn is not 0 the the numerator of An does not converge. But that does not really help. Im trying to figure out which convergence theorem to use........
     
  5. Sep 5, 2011 #4

    SammyS

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    So the n-th term, An, is the average of the first n terms of sequence (Xn).
     
  6. Sep 5, 2011 #5
    right!!! But I am still stuck...
     
  7. Sep 5, 2011 #6

    Dick

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    You need to put together an argument using the definition of limit here. Pick an epsilon, how would find an N such that the average of the terms up to k is within epsilon of L for all k>N? The initial terms in the sequence may not be close to L, the later ones have to be. You need to take both groups into account.
     
    Last edited: Sep 5, 2011
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