1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Limit of a sequence proof

  1. Mar 1, 2006 #1
    I am really stuck with this excercise:
    Let [tex]a_n and b_n[/tex] be two sequences, where [tex]b_n = \frac{a_1 + a_2 + ... + a_n}{n}[/tex] and [tex]lim_(n \rightarrow \infinity) a_n = a[/tex] prove that [tex]lim_(n \rightarrow \infinity) b_n = a[/tex].

    I tried to use the definition - [tex]\mid b_n - a \mid < \frac{\mid a_1 - a \mid + \mid a_2 -a \mid + ... + \mid a_n - a \mid}{n}[/tex], but i dont know how to proceed. Any ideas?
     
  2. jcsd
  3. Mar 12, 2006 #2

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    Okay, try to think about what's going on. You have a sequence an going to a. So for any e > 0, you can choose a sufficiently large N such that an is within e of a, if n > N. So eventually, everything in the sequence gets as close as you want to a. Now bn is the sequence of averages. You want to show that as a sequence approaches a, so does the corresponding sequence of averages. So you want to show that for any e' > 0, you can choose N' sufficiently large so that the average bn is within e of a, for n > N'. Just choose your N' so large that regardless of the ai that are far away from a, there are enough ai, with i < N' close to a so that the average from a1 to aN' is still close to a. If you have infinitely many numbers, and "most" of them are close to a, then you can choose a finite number of them so that even of some of the ones you choose are very far from a, so many of them are close to a that the average is close to a. And of course, if the average is close for N', then it will work for all bn where n > N'. Do you get the idea? If so, then you just need to state it rigorously, and if you get the idea then it shouldn't be hard.
     
  4. Mar 13, 2006 #3

    benorin

    User Avatar
    Homework Helper

    It didn't display when I loaded it, so I fixed it in quote.
     
    Last edited: Mar 13, 2006
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Limit of a sequence proof
  1. Sequence limit proof (Replies: 7)

Loading...