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Telescopic sum issues, cant get Sk

  1. Nov 13, 2015 #1
    1. The problem statement, all variables and given/known data
    Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges

    2. Relevant equations
    3/(1*2*3) + 3,/(2*3*4) + 3/(3*4*5) +......+ 3/n(n+1)(n+2)
    3. The attempt at a solution
    the first try, i tried using partial fraction which equals to A/(n) + B/(n+1) + C/(n+2). which made me get (3/2n - 3/n+1 + 3/2(n+2). Sn= (3/2 - 3/2 +3/6) + (3/4 - 1 + 3/8) + (3/6 - 3/4 + 3/10) + (3/8 - 3/5 + 3/12). i cant cancel it correctly. please help
    i saw the correct working which the partial fraction suppose to be A/(n)(n+1) - B/(n+1)(n+2). didnt make sense to me because i thought in partial fraction we are suppose to split all the parts? and i tried that way and i would get A/(n)(n+1) + B/(n+1)(n+2) instead of A/(n)(n+1) - B/(n+1)(n+2). (couldnt get negative b) please help, have been trying to solve this for days.
     
    Last edited by a moderator: Nov 13, 2015
  2. jcsd
  3. Nov 13, 2015 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Suggestions: (i) forget the '3' in the numerator; you can always put it back after you have done the summation; (ii) write ##\frac{1}{n(n+2)}## in partial fractions, then multiply by ##1/(n+1)## afterward.
     
  4. Nov 13, 2015 #3

    Mark44

    Staff: Mentor

    First off, you need more parentheses. When you write 3/2n, that means ##\frac 3 2 n##, not ##\frac3 {2n}## as you intended. When you write 3/n + 1, that's even worse, as it means ##\frac 3 n + 1##
    You can split it however you want. I wouldn't have thought of this approach, but working it through, it makes sense.

    Decompose by setting ##\frac{1}{n(n + 1)(n + 2)} = \frac{A}{n(n + 1)} + \frac{B}{(n + 1)(n + 2)}##. After you find A and B, you will have a series that telescopes nicely.
    Note: to simplify things you can take out a factor of 3 from all of the terms in the original problem.
     
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