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Series with positive terms

  1. Nov 22, 2006 #1
    Let (infinity)(sigma)(n=1) = An be a series with positive terms such that lim(n -> infinity) = (An+1)/(An) = L < 1

    a) Let L < r < 1. Show that there is an N > 0 such that for all n > N, we have (An+1)/(An) < r

    b) Show that Ak+N < or = ANr^k for k = 1, 2....

    c) Show that lim (k -> infinity) (Ak+N)^(N+k) < or = r


    For An+1, it's A with sub n+1
    An, it's A sub n
    Ak+N is A such (k+N)
    ANr^k is A sub N times r^k

    Last edited by a moderator: May 8, 2014
  2. jcsd
  3. Nov 22, 2006 #2


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    Science Advisor

    Use the DEFINITION of "limit of a sequence".

    Proof by induction on k.

    After b, this should be obvious. What is the limit of rN+k?
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