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Prove an infinite sum exists and its sum

  1. Feb 18, 2016 #1

    B3NR4Y

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    Gold Member

    1. The problem statement, all variables and given/known data
    Let {b k } be a sequence of positive numbers. Assume that there exists a sequence {a k}, such that a k is greater than or equal to 0 for all k, a_k is decreasing, the limit of a_k is 0 and b_k = a_k - a _(k+1). Show that the sum from k=1 to infinity of b k exists and equals a_1

    2. Relevant equations
    Not really any I can think of

    3. The attempt at a solution
    I'm not sure how to prove this. The sum of b_k has two parts that both go to zero, but I can think of an a_k that satisfies all those properties but doesn't converge (1/k), but if you write out the sum you can clearly see all the terms cancel except the a_1 term.

    For example:

    Sum of b_k = (a1 - a 2 ) + (a_2-a_3)+(a_3-a_4)+.....

    Which clearly cancels all the terms except a_1. But this doesn't seem rigoruous enough.


    (Also sorry for inconsistent subscripts, I'm on my phone)
     
  2. jcsd
  3. Feb 18, 2016 #2

    RUber

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    Homework Helper

    I would start by doing just what you did. Compute the partial sum,
    ##\sum_{i=1}^{N-1} b_k##
    Then show that in the limit as N goes to infinity, you get what you expect.
     
  4. Feb 18, 2016 #3

    Ray Vickson

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    Don't forget that the infinite sum ##\sum_{i=1}^{\infty} b_i ## is defined as the limit ##\lim_{n \to \infty} \sum_{i=1}^n b_i##. That is, the infinite sum exists if and only if the limit of finite sums exist.
     
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