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Math Help - Limit of 2 sequences

  1. #1
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    Limit of 2 sequences

    Hi,

    I'm having some trouble with this question:

    Show that if {a_{n}} converges to L, that {b_{n}} where  b_{n}=\frac{1}{n}\left(a_{1}+...+a_{n}\right) also converges to L. Look at the case L=0 first.

    I got as far as saying that

    |\frac{1}{n}\left(a_{1}+...+a_{n}\right)-0|<\epsilon

    |\frac{1}{n}\left(a_{1}+...+a_{n}\right)|\leq\frac  {1}{n}\left(|a_{1}|+...+|a_{n}|\right)

    and the right part of the inequality can be rewritten as

    \frac{1}{n}\sum^{N}_{i=1}|a_{i}|+\frac{1}{n}\sum^{  n}_{i=N}|a_{i}|

    How should I continue?

    Thank you
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  2. #2
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    Quote Originally Posted by mikado View Post
    Show that if {a_{n}} converges to L, that {b_{n}} where  b_{n}=\frac{1}{n}\left(a_{1}+...+a_{n}\right) also converges to L. Look at the case L=0 first.
    \frac{1}{n}\left( {\sum\limits_{k = 1}^n {a_k } } \right) - L = \sum\limits_{k = 1}^n {\frac{{a_k  - L}}{n}}  = \sum\limits_{k = 1}^N {\frac{{a_k  - L}}{n}}  + \sum\limits_{k = N + 1}^n {\frac{{a_k  - L}}{n}} .

    If \varepsilon  > 0\, \Rightarrow \,\left( {\exists N} \right)\left[ {n \geqslant N\, \Rightarrow \,\left| {a_n  - L} \right| < \frac{\varepsilon }{2}} \right].

    Now \,\left( {\exists K > N} \right)\left[ {\frac{1}{K}\left( {\sum\limits_{k = 1}^N {\frac{{a_k  - L}}{n}} } \right) < \frac{\varepsilon }{2}} \right].

    Carry on.
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