1. Not finding help here? Sign up for a free 30min 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!

Question with Summing a Series (Non-Geometric)

  1. Oct 1, 2009 #1
    1. The problem statement, all variables and given/known data
    [tex]\stackrel{infinity}{n=1}\sum\left(\frac{2}{n^{2}+8n+15}\right)[/tex]


    2. Relevant equations
    Partial Sums
    Knowledge of Series

    3. The attempt at a solution
    Please see the attached word document for previous work up until this part. Also, please excuse the improper LaTeX usage [I'm getting better]!

    To summarize:
    [tex]\stackrel{infinity}{n=1}\sum\left(\frac{2}{n^{2}+8n+15}\right)[/tex]
    = [tex]\stackrel{infinity}{n=1}\sum\left(\frac{1}{n+3}-\frac{1}{n+5}\right)[/tex]

    Now, when I string out the sum I start to have concept issues that neither my book, my lecture notes, nor web searches have been able to explain:
    [tex]\left(\frac{1}{4}-\frac{1}{6}\right)+\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{6}-\frac{1}{8}\right) + ... + \left(\frac{1}{n}-\frac{1}{n+2}\right) + \left(\frac{1}{n+1}-\frac{1}{n+3}\right) + \left(\frac{1}{n+2}-\frac{1}{n+4}\right) + \left(\frac{1}{n+3}-\frac{1}{n+5}\right) [/tex]

    I think the variable increments are correct at the end, this is one of my questions...

    However, another - important - question is which variable increments cancel?

    I understand that only:
    [tex]\left(\frac{1}{4} + \frac{1}{5}\right) + ... [/tex]

    Will remain since those particular terms are lower than the subtracting part of the sum, however, I'm unsure about where to start with canceling variable terms.

    Here is my 'guess' on which variable terms will remain:
    [tex] ... + \left(-\frac{1}{n+4}-\frac{1}{n+5}\right) [/tex]
    This was done under the assumption that the positive part of the sum will never reach these particular terms.

    From there take the limit of the Series:

    [tex]lim_{n->infinity} \left(\frac{1}{4} + \frac{1}{5} -\frac{1}{n+4}-\frac{1}{n+5}\right)[/tex]

    [tex]lim_{n->infinity} \left(\frac{5}{20} + \frac{4}{20} - 0 - 0 \right)[/tex]

    [tex]lim_{n->infinity} \left(\frac{9}{20} \right)[/tex]


    Sincerely,

    NastyAccident
     

    Attached Files:

  2. jcsd
  3. Oct 1, 2009 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    If you're unsure which terms cancel, write out the first twelve terms and see which ones cancel
     
  4. Oct 1, 2009 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, you stopped one term too soon! What is the term immediately after [itex]\left(\frac{1}{6}- \frac{1}{8}\right)[/itex]? So you see now what cancels? When isn+ 3= m+ 5?
     
  5. Oct 1, 2009 #4
    Ahh, thanks! So, pretty much anything that n+3=n+5 will cancel (on both the beginning and end).

    Got it now! Thanks for clearing that blip up.

    Sincerely,

    NastyAccident
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Question with Summing a Series (Non-Geometric)
  1. Sum of geometric series (Replies: 10)

Loading...