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Math Help - if {a(n)} converges to a, (1/n)(sum k=1, n of {a(k)}) converges to a

  1. #1
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    Angry if {a(n)} converges to a, (1/n)(sum k=1, n of {a(k)}) converges to a

    i have to prove the following,

    if {a(n)} converges to a, (1/n)(sum k=1, n of {a(k)}) converges to a
    can you help me? :/
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  2. #2
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    Re: if {a(n)} converges to a, (1/n)(sum k=1, n of {a(k)}) converges to a

    Quote Originally Posted by nappysnake View Post
    i have to prove the following,
    if {a(n)} converges to a, (1/n)(sum k=1, n of {a(k)}) converges to a
    There are two cases.
    Case I, Suppose that a=0. Notation {M_n} = \sum\limits_{k = 1}^n {\frac{{{a_k}}}{n}} and {S_n} = \sum\limits_{k = 1}^n {{a_k}}.

    Then \left| {{M_n}} \right| = \left| {\frac{{{S_n}}}{n}} \right| \leqslant \left| {\frac{{{S_n} - {S_K} + {S_K}}}{n}} \right| \leqslant \frac{1}{n}\sum\limits_{k = K + 1}^n {\left| {{a_k}} \right|}  + \frac{{\left| {{S_K}} \right|}}{n}
    Now I leave the details to you.


    Case II Suppose a\ne 0.
    Then if b_n=a-a_n then note that (b_n)\to 0.

    Fall back to case I.
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