Results 1 to 3 of 3

Math Help - Integral

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    62

    Integral

    HI,

    I must show that:  (\forall n\in \mathbb{N}) \sum_{k=0}^n \frac{(-1)^kx^k}{k!}+(-1)^{n+1}\int_0^{x} \frac{(x-t)^n}{n!}e^{-t} dt= e^{-x}

    I find that the expression is true for n=0; but i don't know how to continue?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Black's Avatar
    Joined
    Nov 2009
    Posts
    105
    It's enough to show that

    (-1)^{n+1}\int_{0}^{x}\frac{(x-t)^n}{n!}e^{-t}dt=\sum_{k\,=\,n+1}^{\infty}\frac{(-1)^kx^k}{k!}.

    We have

    (-1)^{n+1}\int_{0}^{x}\frac{(x-t)^n}{n!}e^{-t}dt=\frac{(-1)^{n+1}}{n!}\int_{0}^{x}(x-t)^n \left(1-t+\frac{t^2}{2!}-\frac{t^3}{3!}+\cdots \right)dt.

    Now use integration by parts to show that

    \int_{0}^{x}\frac{t^j}{j!}(x-t)^ndt=\frac{x^{n+j+1}}{(n+1)(n+2) \cdots(n+j+1)} for j=0,1,2,\dots.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by lehder View Post
    HI,

    I must show that:  (\forall n\in \mathbb{N}) \sum_{k=0}^n \frac{(-1)^kx^k}{k!}+(-1)^{n+1}\int_0^{x} \frac{(x-t)^n}{n!}e^{-t} dt= e^{-x}

    I find that the expression is true for n=0; but i don't know how to continue?

    Try induction on n: for n = 0 you already did, so now suppose it's true for all k up to n and try to prove for k = n + 1:

    Put I_n:=\sum_{k=0}^n \frac{(-1)^kx^k}{k!}+(-1)^{n+1}\int_0^{x} \frac{(x-t)^n}{n!}e^{-t} dt , so doing integration by parts with u:=(x-t)^{n+1}\,,\,\,v':=e^{-t} , we get:

    I_{n+1}:= \sum_{k=0}^{n+1} \frac{(-1)^kx^k}{k!}+(-1)^{n+2}\int_0^{x} \frac{(x-t)^{n+1}}{(n+1)!}e^{-t} dt =\frac{(-1)^{n+1}x^{n+1}}{(n+1)!}+\sum\limits_{k=0}^n\frac{  (-1)^nx^k}{k!}+ \frac{(-1)^{n+2}}{(n+1)!}\left\{\left[-(x-t)^{n+1}e^{-t}\right]_0^x-(n+1)\int\limits_0^x (x-t)^ne^{-t}dt\right\}=

    =\frac{(-1)^{n+1}x^{n+1}}{(n+1)!}+\sum\limits_{k=0}^n\frac{  (-1)^nx^k}{k!}+\frac{(-1)^{n+2}x^{n+1}}{(n+1)!}+(-1)^{n+3}\int\limits_0^x\frac{(x-t)^n}{n!}e^{-t}\,dt .

    Now just check that you can cancel out the first and the third summands, and the second and fourth ones are exactly the same as I_n and you're done.

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: August 31st 2010, 08:38 AM
  2. Replies: 1
    Last Post: June 2nd 2010, 03:25 AM
  3. Replies: 0
    Last Post: May 9th 2010, 02:52 PM
  4. Replies: 0
    Last Post: September 10th 2008, 08:53 PM
  5. Replies: 6
    Last Post: May 18th 2008, 07:37 AM

Search Tags


/mathhelpforum @mathhelpforum