Results 1 to 4 of 4

Math Help - Deriving the series of e using binomial theorem

  1. #1
    Newbie
    Joined
    Jun 2010
    Posts
    2

    Deriving the series of e using binomial theorem

    I need help understanding how the series of e derives into the exponential series using the binomial theorem.

    Here is a link to a pic of a page in my book, regarding the exponential series:

    http://i46.tinypic.com/qz0oat.jpg

    A couple of questions:

    Where does the [1 + (1/k)]^k come from and why is it used?

    Could you clarify the expansion of [1+(1/k)]^k?
    I don't understand how it gets to ... k(1/k) + k(k-1)/2! (1/k^2) + ...

    How does it end up with a 1 + 1 + 1[1-(1/k)]/2! + ...

    Why are you finding the limit of the series?

    And finally how do you end up with exponential series

    x^n /n! = 1 + x + x^2/2! + ... ?

    I'm confused and just really don't understand why or how you end up with everything. Try and keep it simple, please. Help is VERY appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,656
    Thanks
    1480
    Quote Originally Posted by lobbyistboy View Post
    I need help understanding how the series of e derives into the exponential series using the binomial theorem.

    Here is a link to a pic of a page in my book, regarding the exponential series:

    http://i46.tinypic.com/qz0oat.jpg

    A couple of questions:

    Where does the [1 + (1/k)]^k come from and why is it used?

    Could you clarify the expansion of [1+(1/k)]^k?
    I don't understand how it gets to ... k(1/k) + k(k-1)/2! (1/k^2) + ...

    How does it end up with a 1 + 1 + 1[1-(1/k)]/2! + ...

    Why are you finding the limit of the series?

    And finally how do you end up with exponential series

    x^n /n! = 1 + x + x^2/2! + ... ?

    I'm confused and just really don't understand why or how you end up with everything. Try and keep it simple, please. Help is VERY appreciated.
    Remember that the exponential function is such that

    f'(x) = f(x).


    Suppose you wanted to find the derivative of a^x.


    Then f'(x) = \lim_{h \to 0}\frac{a^{x + h} - a^x}{h}

     = \lim_{h \to 0}\frac{a^xa^h - a^x}{h}

     = \lim_{h \to 0}\frac{a^x(a^h - 1)}{h}

     = a^x\lim_{h \to 0}\frac{a^h - 1}{h}.


    Now, if you wanted to find the value of a that makes f'(x) = f(x), you need to let \lim_{h \to 0}\frac{a^h - 1}{h} = 1 and solve for a.


    Then \lim_{h \to 0}\frac{a^h - 1}{h} = 1

    \lim_{h \to 0}(a^h - 1) = \lim_{h \to 0}h

    \lim_{h \to 0}a^h = \lim_{h \to 0}(1 + h)

    \lim_{h \to 0}(a^h)^{\frac{1}{h}} = \lim_{h \to 0}(1 + h)^{\frac{1}{h}}

    a = \lim_{h \to 0}(1 + h)^{\frac{1}{h}}.


    Now suppose h = \frac{1}{k}. Then as h \to 0, k \to \infty, so

    a = \lim_{k \to \infty}\left(1 + \frac{1}{k}\right)^k.


    Now we just give a the name Euler's Number e, since Euler was the one who discovered it, and we can finally say

    e = \lim_{k \to \infty}\left(1 + \frac{1}{k}\right)^k.



    The rest involves using the Binomial theorem to expand \left(1 + \frac{1}{k}\right)^k and letting k \to \infty.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2010
    Posts
    2
    Thanks, that was extremely helpful. I kind of understand how it was derived.

    But one part that still confuses me is how the expanded form of [1+(1/k)]^k simplifies to:

    1 + 1 + 1[1-(1/k)]/2! + ...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,656
    Thanks
    1480
    Quote Originally Posted by lobbyistboy View Post
    Thanks, that was extremely helpful. I kind of understand how it was derived.

    But one part that still confuses me is how the expanded form of [1+(1/k)]^k simplifies to:

    1 + 1 + 1[1-(1/k)]/2! + ...
    That's just an application of the Binomial Theorem.

    (a + b)^n = \sum_{r = 0}^n{{n\choose{r}}a^{n-r}b^r}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Deriving Sum from Fourier Series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 4th 2011, 07:34 AM
  2. Deriving heat diffusion equation with Gauss' Divergence Theorem?
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: November 25th 2009, 04:06 PM
  3. Series and Sequences..deriving the formula
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: November 5th 2009, 06:40 AM
  4. Binomial Theorem or Binomial Coefficient
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: October 2nd 2009, 01:06 PM
  5. [SOLVED] Binomial Theorem(Sum of series)?
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 23rd 2009, 06:18 AM

Search Tags


/mathhelpforum @mathhelpforum