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!

Power series

  1. May 7, 2006 #1
    Find the sum of the series:
    [tex]\sum\limits_{n = 1}^\infty {nx^n } [/tex] if [tex]
    \left| x \right| < 1

    [/tex]


    I thought maybe with the geometric form, but im not sure.
     
    Last edited: May 7, 2006
  2. jcsd
  3. May 7, 2006 #2
    Is it asking for a number, or just if the series converges?
     
  4. May 7, 2006 #3
    I think for a general solution. It should converge.
     
  5. May 7, 2006 #4
    Is it asking you to find a power series representation??
     
  6. May 7, 2006 #5
    It already is a power series... It's asking for an expression for the sum.
     
  7. May 7, 2006 #6
    You might start by writing out partial sums and see if that gets you anywhere.....
     
  8. May 7, 2006 #7

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    How does this differ from your usual geometric series?
     
  9. May 7, 2006 #8

    Curious3141

    User Avatar
    Homework Helper

    Hint : Call the original series S. Write out the first five or so terms in the series. Divide the series by x to get a new series (S/x). Now take the difference between this new series and the original series (S/x - S), term by term and see what you end up with.

    The other way to do it is to differentiate a geometric series, but that's more complicated and unnecessary.
     
    Last edited: May 7, 2006
  10. May 8, 2006 #9

    Pyrrhus

    User Avatar
    Homework Helper

  11. May 8, 2006 #10

    Curious3141

    User Avatar
    Homework Helper

    Comparing series like these to derivatives of geometric series is a nice and interesting approach (I used to do this), but in most cases I've found that simply dividing or multiplying by x is an easier approach. :smile:
     
  12. May 8, 2006 #11

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    May as well have a third approach:

    [tex]\sum_{n=1}^{\infty}nx^n=\sum_{n=1}^{\infty}\sum_{i=1}^{n}x^n[/tex]

    Change the order of summation (absolutely convergent series) then apply geometric series a couple of times. This is maybe the most complicated of the three, practice in rearranging summations never hurt.
     
    Last edited: May 8, 2006
  13. May 8, 2006 #12
    thanks for all the hints.

    The method I had to use is the derivative of the geometric series (similar to the one used for the maclaurin problem) using

    [tex]

    \left( {\frac{1}{{1 - x}}} \right)^\prime = \sum\limits_{n = 0}^\infty {nx^{n - 1} }


    [/tex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Power series
  1. Power series (Replies: 5)

  2. Power series (Replies: 3)

  3. Power Series (Replies: 13)

  4. Power series (Replies: 11)

  5. Power Series (Replies: 2)

Loading...