Results 1 to 3 of 3

Math Help - The principle of Taylor McLaurin

  1. #1
    Newbie
    Joined
    Dec 2008
    Posts
    1

    The principle of Taylor McLaurin

    Please I need an explanation of what it is all about and how it is used to prove that: cosx plus sinx equals e^ix
    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,621
    Thanks
    426
    \cos{x} + i\sin{x} = e^{ix}

    here's a good explanation ...

    Fermat's Last Theorem: Euler's Formula
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1

    Maclaurin Series and Euler's Formula

    Hi -

    The general idea behind Maclaurin Series is to show you how to write functions like sin x, cos x and e^x as an infinite power series in x. (The Taylor series is similar, but is a bit more complicated. In fact, the Maclaurin is a special case of the Taylor series.)

    If you don't know what I mean by an infinite power series, you're probably best to start by finding out about Infinite Geometric Series first. This will help you to understand how an infinite series can have a finite (and useful!) sum.

    It's too complicated to show you here how the Maclaurin series is derived and used, but if you look at the .pdf file at http://mathinsite.bmth.ac.uk/pdf/macseries_theory.pdf you'll find a thorough explanation.

    You need to look in particular at:

    • Page 4, where you'll find the formula for the Maclaurin Series itself, and how it gives you the series for sin x and e^x
    • Page 6, where you'll find the series for cos x

    To use these to prove Euler's formula, cos x + i sin x = e^{ix} is very straightforward:

    • Multiply both sides of the sinx series by i, noting that i^3 = -i, i^5=i, etc.
    • Add the result to the series for cosx, arranging the terms in ascending powers of x.
    • In the series expansion of e^x, replace x by ix. Note this time that i^2=-1, i^4=1, etc.
    • Compare the result with what you found for cosx+isinx. You should find they're equal.

    There's a proof in the Wikipedia article at Euler's formula - Wikipedia, the free encyclopedia.

    Hope that helps.
    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. mclaurin series and taylor inequality
    Posted in the Calculus Forum
    Replies: 6
    Last Post: May 10th 2010, 03:05 AM
  2. McLaurin series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 2nd 2010, 02:14 AM
  3. McLaurin Series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 20th 2009, 06:17 PM
  4. McLaurin Expansion
    Posted in the Calculus Forum
    Replies: 6
    Last Post: April 24th 2008, 01:17 PM
  5. Mclaurin/Taylor series expansion
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 21st 2006, 01:39 PM

Search Tags


/mathhelpforum @mathhelpforum