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Summation of series

  1. Feb 7, 2013 #1
    1. The problem statement, all variables and given/known data

    Let v1, v2, v3 be a sequence and let

    un=nvn-(n+1)vn+1

    for n= 1,2,3.... find [itex]\sumun[/itex] from n=1 to N.
    2. Relevant equations


    3. The attempt at a solution
    Began with method of differences and arrived at
    Sn= v1-(n+1)vn+1
     
  2. jcsd
  3. Feb 7, 2013 #2

    jedishrfu

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    I'd first try writing out u1, u2, u3 to see if there's some term cancellations that come about when you sompute
    sum (un) = u1 + u2 + u3 + ...
     
  4. Feb 8, 2013 #3

    SammyS

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    What's your question?
     
  5. Feb 8, 2013 #4
    I think the question is: given the recursive formula ##u_n = nv_n - (n+1)v_{n+1}##, find the general formula for the sum of ##u_1 + u_2 + ... + u_n## for any n.

    Let's first list out some possibilities:

    ##u_1 = v_1 - 2v_2\\ u_2 = 2v_2 - 3v_3\\ u_3 = 3v_3 - 4v_4##

    So the sum of the three is:

    sum{##u_3##} ##= v_1 - 2v_2 + 2v_2 - 3v_3 + 3v_3 - 4v_4 = v_1 - 4v_4##

    Based on this, can you think of a formula for any ##n##th sum?
     
    Last edited: Feb 8, 2013
  6. Feb 9, 2013 #5

    SammyS

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    If you mean [itex]\displaystyle\ \ S_N=\sum_{n=1}^{N}u_n=v_1-(N+1)v_{N+1},,\ [/itex] then your result looks good.
     
    Last edited: Feb 9, 2013
  7. Feb 9, 2013 #6

    jedishrfu

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    Do you have one too many equals?
     
  8. Feb 9, 2013 #7

    SammyS

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    LOL !

    Thanks!

    Actually I had one too few un .

    I'll edit my post!
     
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