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Math Help - Proving a Bernoulli Relationship

  1. #1
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    Proving a Bernoulli Relationship

    I have already managed to prove the relationship
    Proving a Bernoulli Relationship-1.jpg

    By considering the integral

    Proving a Bernoulli Relationship-integral.jpg
    I am now trying to prove
    Proving a Bernoulli Relationship-2.jpg

    This is what I have managed

    to do so far but now I have no idea where to move forward
    Proving a Bernoulli Relationship-3.jpg
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Proving a Bernoulli Relationship

    Quote Originally Posted by klw289 View Post
    I have already managed to prove the relationship
    Click image for larger version. 

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    By considering the integral

    Click image for larger version. 

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    I am now trying to prove
    Click image for larger version. 

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    This is what I have managed

    to do so far but now I have no idea where to move forward
    Click image for larger version. 

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    You can start from the 'explicit definition' of the 'Bernoulli polynomial of index n'...

    B_{n}(x)= \sum_{k=0}^{n} \binom{n}{k}\ B_{k}\ x^{n-k} (1)

    ... where B_{k} isw the 'Bernoulli number of index k'. Deriving (1) You obtain...

    B_{n}^{'}(x)= \sum_{k=0}^{n} \binom{n}{k}\ B_{k}\ (n-k)\ x^{n-k-1}=

    = n\ \sum_{k=0}^{n-1} \binom{n-1}{k}\ B_{k}\ x^{n-k-1} = n\ B_{n-1}(x) (2)

    Kind regards

    \chi \sigma
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  3. #3
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    Re: Proving a Bernoulli Relationship

    I've have already proved that relationship as I mentioned, I used it to help me integrate the intergral I was asked to consider but how do I use these results to get to the proof for Bn+1 which I am looking for
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