Results 1 to 2 of 2

Thread: Limit of 2 sequences

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    3

    Limit of 2 sequences

    Hi,

    I'm having some trouble with this question:

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

    I got as far as saying that

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

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

    and the right part of the inequality can be rewritten as

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

    How should I continue?

    Thank you
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,776
    Thanks
    2823
    Awards
    1
    Quote Originally Posted by mikado View Post
    Show that if $\displaystyle {a_{n}}$ converges to L, that $\displaystyle {b_{n}}$ where $\displaystyle b_{n}=\frac{1}{n}\left(a_{1}+...+a_{n}\right)$ also converges to L. Look at the case L=0 first.
    $\displaystyle \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 $\displaystyle \varepsilon > 0\, \Rightarrow \,\left( {\exists N} \right)\left[ {n \geqslant N\, \Rightarrow \,\left| {a_n - L} \right| < \frac{\varepsilon }{2}} \right]$.

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

    Carry on.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. use sequences to prove limit of |f(x)| HELP!
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Nov 14th 2009, 04:40 PM
  2. How do you find the limit of these sequences?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Oct 29th 2009, 06:54 PM
  3. Limit of the quotient of two sequences
    Posted in the Math Challenge Problems Forum
    Replies: 11
    Last Post: Aug 25th 2009, 05:27 PM
  4. Limit Theorems, Sequences
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: Mar 30th 2009, 05:30 PM
  5. Limit of Sequences
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jan 1st 2008, 11:23 AM

Search Tags


/mathhelpforum @mathhelpforum