Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding closed form for the next series:

  1. Mar 30, 2007 #1
    i need to find the closed form expressiosn for the next sums:
    [tex]g(x)=\sum_{n=0}^{\infty}\frac {x^{2n}}{(2n)!}[/tex]
    and [tex]f(x)=\sum_{n=0}^{\infty} \frac {x^{4n+1}}{4n+1}[/tex]
    well for the second one i thought differentiating, i.e [tex]f'(x)= \sum_{n=0}^{\infty} x^{4n}=1/(1-x^4)=\frac {1}{(1+x^2)*(1-x^2)}[/tex], now i could integrate it by parts but in my course i havent yet got to integrals, so i cannot use integration by parts, do you have any other way to compute this sum.
    for the first one, it looks like the power series of e^x, i tried to recursively get to the expression but so far to no success.
    any pointers would be helpful.
     
    Last edited: Mar 30, 2007
  2. jcsd
  3. Mar 30, 2007 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The first one is, in fact,
    [tex]cosh(x)= \frac{e^x+ e^{-x}}{2}[/itex]

    For the second one, your idea of differentiating to get a geometric series and then integrating works nicely (after you learn to integrate, of course!) using partial fractions, not "by parts". But it gives a function involving arctan and logarithms so I doubt you will find any simple way to do that.
     
  4. Mar 30, 2007 #3

    Gib Z

    User Avatar
    Homework Helper

    Just in case you wanted it, the second ones

    [tex]\frac{1}{4} (2\arctan x -\ln (x^2-1))[/tex] I think.
     
  5. Mar 30, 2007 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    The summand for g(x) looks similar to the summand for e^x -- this suggests you might be able to express the power series for g(x) in terms of the power series for e^x.

    The summand for f(x) looks similar to the summands for log(1+x) and for arctan x -- this suggests you might be able to express the power series for g(x) in terms of the power series for log(1+x) and arctan x.
     
  6. Mar 31, 2007 #5
    after one hour i got it by myself.
    but i appreciate your help.
    btw, iv'e got something like this:
    i need to find a power series for (1+x^2)*arctg(x).
    i know the power series of arctg(x), but when im multiplying this sum with (1+x^2), i get x+sum, i.e where the sum starts from n=1, but i cant put the x into the sum. perhaps it should be this way?
     
  7. Mar 31, 2007 #6

    Gib Z

    User Avatar
    Homework Helper

    What is arctg(x)????
     
  8. Mar 31, 2007 #7
    the same as arctan(x).
    it's one of those shorcuts,like for sinh, there's sh, etc.
     
  9. Mar 31, 2007 #8

    Gib Z

    User Avatar
    Homework Helper

    So are you saying arctg(x) is the same as arctan(x)? Or that its a similar pattern, its the inverse of g(x)? Because thats argcosh(x).
     
  10. Mar 31, 2007 #9
    yes arctg(x) is the same as arctan(x), now back on topic.
     
  11. Mar 31, 2007 #10

    Gib Z

    User Avatar
    Homework Helper

    Well then yes, it can be the way you stated, it doesn't matter and isnt wrong.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Finding closed form for the next series:
Loading...