Results 1 to 4 of 4

Math Help - Finding terms for maclaurin series

  1. #1
    dyy
    dyy is offline
    Newbie
    Joined
    May 2007
    Posts
    2

    Question Finding terms for maclaurin series

    Problem- Find the terms through x^5 in the Maclaurin series for f(x)=e^(-x) * cos(x)

    After trying to find f'(x), f''(x), f'''(x), and so on, the derivatives seemed overly complex. I heard that it is possible to use known Maclaurin series (e.g. e^(-x) and cos(x) separately) and then perform multiplications, divisions, etc. How would I combine them?

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by dyy View Post
    Problem- Find the terms through x^5 in the Maclaurin series for f(x)=e^(-x) * cos(x)

    After trying to find f'(x), f''(x), f'''(x), and so on, the derivatives seemed overly complex. I heard that it is possible to use known Maclaurin series (e.g. e^(-x) and cos(x) separately) and then perform multiplications, divisions, etc. How would I combine them?

    Thanks!
    Yes.

    e^x = 1 + x + \frac{x^2}{2!} + ...
    So,
    e^{-x} = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+...

    And

    \cos x = 1 - \frac{x^2}{2!}+ \frac{x^4}{4!}-...

    That means,
    e^{-x} \cos x = \left( 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+... \right)\left( 1 - \frac{x^2}{2!}+ \frac{x^4}{4!}-... \right)

    Now preform something called the Cauchy Product.
    On the first 5 terms in each one.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by dyy View Post
    Problem- Find the terms through x^5 in the Maclaurin series for f(x)=e^(-x) * cos(x)

    After trying to find f'(x), f''(x), f'''(x), and so on, the derivatives seemed overly complex. I heard that it is possible to use known Maclaurin series (e.g. e^(-x) and cos(x) separately) and then perform multiplications, divisions, etc. How would I combine them?

    Thanks!
    Another way of doing this is to write:

    <br />
f(x)= Re\left[ e^{x(-1+i)}\right]<br />

    Then:

    <br />
f(x)= Re\left[<br />
1 + [x(-1+i)] + \frac{[x(-1+i)]^2}{2!} + \frac{[x(-1+i)]^3}{3!}+\frac{[x(-1+i)]^4}{4!}+ ...<br />
\right]<br />

    The right hand side of this is fairly easy to simplify to find the real part
    and so find the required series.

    RonL
    Follow Math Help Forum on Facebook and Google+

  4. #4
    dyy
    dyy is offline
    Newbie
    Joined
    May 2007
    Posts
    2
    I'm not sure I understand the cauchy product entirely unfortunately-

    I'm trying to multiply the terms from the two series- so far I have-

    1+(-1)x+(1/2-1/2)x^2+[the third term]<-where I'm stuck, since there's no term in cosx with x^3. What do I multiply the x^3/3! term from e^(-x) with, if anything?

    Thanks once again.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: June 2nd 2010, 09:40 AM
  2. Finding a limit. Finding Maclaurin series.
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 18th 2010, 10:04 PM
  3. Finding a limit using Maclaurin Series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 2nd 2009, 01:23 AM
  4. Replies: 1
    Last Post: August 8th 2008, 02:25 PM
  5. Finding Maclaurin Series By Division
    Posted in the Calculus Forum
    Replies: 6
    Last Post: May 3rd 2008, 03:29 PM

Search Tags


/mathhelpforum @mathhelpforum