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

  1. Apr 14, 2010 #1
    1. The problem statement, all variables and given/known data
    [PLAIN]http://img263.imageshack.us/img263/9336/seriesgay.jpg [Broken]

    In the previous part of the question we had to show where the taylor expansion comes from, and calculated the maclaurin series for e^x, sin x and cos x. From that we had to prove De Moivre's theorem and so I would imagine that these things help in the last part of this question. I can see it looks like a Maclaurin series, just not sure where to start really.

    Any help would be appreciated, thanks.
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Apr 14, 2010 #2
    What have you tried? Does that infinite series sort of look like any other infinite series you know?
  4. Apr 14, 2010 #3
    well it looks like a maclaurin series, but I don't really know how to work out what it's a Maclaurin series of.
  5. Apr 14, 2010 #4


    Staff: Mentor

    Not really. A Maclaurin series has powers of x (or whatever the variable happens to be).

    IOW, a Maclaurin series looks like this:
    [tex]\sum_{n = 0}^{\infty} a_n x^n[/tex]

    Your series is
    [tex]\sum_{n = 0}^{\infty} \frac{2^n~cos(n\theta)}{n!}[/tex]
  6. Apr 14, 2010 #5
    Yeah but I was thinking that it looked like something to do with e^(i[tex]\theta[/tex]) to the power of n which would give terms of cos(n[tex]\theta[/tex]). I'm not sure, if not how do I go about tackling the problem?
  7. Apr 14, 2010 #6
    Hint: The real part of exp(i x) = ???
  8. Apr 14, 2010 #7
    cos(x)... sorry it's been a long day! Still not sure how to use that to get any further.
  9. Apr 14, 2010 #8
    Replace all the cos(nx) by exp(i n x) in the summation and then take the real part of the summation.
  10. Apr 14, 2010 #9
    ok, I think I have it.

    Is it the Maclaurin series for


    that seems to work I think :s, meaning that the sum is just what's written above right?
  11. Apr 14, 2010 #10
    or rather:

  12. Apr 14, 2010 #11
    Yes, and now you can simply this using Euler's formula

    exp(ix) = cos(x) + i sin(x)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook