1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Sum of a series question

  1. Mar 23, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the sum of the series.

    [tex] \sum_{k=0}^\infty \frac{1}{(k+1)(k+3)} [/tex]


    2. Relevant equations



    3. The attempt at a solution
    [tex]
    = \frac{1}{1\cdot3} + \frac{1}{2\cdot4} + \frac{1}{3\cdot5} + ... + \frac{1}{(n+1)\cdot(n+3)}
    [/tex]

    [tex]
    = \frac{1}{2} [(1-\frac{1}{3}) + (\frac{1}{2} - \frac{1}{4}) + (\frac{1}{3} - \frac{1}{5}) + ... + (\frac{1}{(n+1)} - \frac{1}{(n+3)})
    [/tex]

    [tex]
    = \frac{1}{2}[ 1 + (\frac{1}{2} + \frac{1}{3} + \frac{1}{n+1}) - (\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... + \frac{1}{n+3}) ]
    [/tex]

    So here
    [tex]
    (\frac{1}{2} + \frac{1}{3} + \frac{1}{n+1}) \to 1
    [/tex]

    [tex]
    (\frac{1}{3} + \frac{1}{4} + \frac{1}{5} + ... + \frac{1}{n+3}) \to 0
    [/tex]

    Then the whole thing sums to 1?
     
  2. jcsd
  3. Mar 23, 2010 #2
    Ok, things went very bad from the second to the third line in your argument; why don't you try to cancel the terms with opposite signs, instead of grouping them?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook