Results 1 to 2 of 2

Math Help - Integral with Recursion Relation

  1. #1
    Member
    Joined
    Jul 2008
    Posts
    119

    Integral with Recursion Relation

    Problem
    ====================
    Let  E_{n} = \int_0^1 x^ne^{x-1}dx

    with the following recursion relation

     E_n = 1 - nE_{n-1}

    Start with the index n = 1 and group to the index n = 20. How can you tell that the results are incorrect and at what index n?

    Attempt
    ====================

    I do not think that the results can be incorrect; I think it is true for all indices.

    First, I integrate  E_n using integration by parts (I forgot how to do the "evaluate from/to" symbol in Latex so I used a bracket)

    E_n = \int_0^1 x^ne^{x-1}dx =  [x^ne^{x-1}]_{0}^{1}  - \int_0^1 nx^{n-1}e^{x-1}dx

    Simplify to get

     = 1- n \int_0^1 x^{n-1}e^{x-1}dx

    Since  E_{n} = \int_0^1 x^ne^{x-1}dx , then  E_{n-1} = \int_0^1 x^{n-1}e^{x-1}dx

    Thus,

     = 1 - nE_{n-1}

    Shouldn't this be true for all indices n? I wrote a MATLAB script that computed  E_n, E_{n-1} at all indices and the results look fine.

    Thank you for reading.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    May 2006
    Posts
    244
    Quote Originally Posted by Paperwings View Post
    Problem
    ====================
    Let  E_{n} = \int_0^1 x^ne^{x-1}dx

    with the following recursion relation

     E_n = 1 - nE_{n-1}

    Start with the index n = 1 and group to the index n = 20. How can you tell that the results are incorrect and at what index n?

    Attempt
    ====================

    I do not think that the results can be incorrect; I think it is true for all indices.

    First, I integrate  E_n using integration by parts (I forgot how to do the "evaluate from/to" symbol in Latex so I used a bracket)

    E_n = \int_0^1 x^ne^{x-1}dx = [x^ne^{x-1}]_{0}^{1} - \int_0^1 nx^{n-1}e^{x-1}dx

    Simplify to get

     = 1- n \int_0^1 x^{n-1}e^{x-1}dx

    Since  E_{n} = \int_0^1 x^ne^{x-1}dx , then  E_{n-1} = \int_0^1 x^{n-1}e^{x-1}dx

    Thus,

     = 1 - nE_{n-1}

    Shouldn't this be true for all indices n? I wrote a MATLAB script that computed  E_n, E_{n-1} at all indices and the results look fine.

    Thank you for reading.
    Are any of them negative? Can such an integral be negative?

    (consider the build up in errors with itterations)
    .
    Last edited by Constatine11; February 2nd 2009 at 11:15 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: April 6th 2011, 11:46 PM
  2. Replies: 1
    Last Post: March 1st 2010, 07:24 AM
  3. Proof of a definite multiple integral relation
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 10th 2010, 04:30 AM
  4. How to prove that integral relation?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 2nd 2008, 03:20 AM
  5. Recursion
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: September 16th 2008, 06:24 PM

Search Tags


/mathhelpforum @mathhelpforum