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Math Help - Zeta function and bernouli numbers proof

  1. #1
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    Zeta function and bernouli numbers proof

    Prove that:


    lim_m _\rightarrow _\infty        \mid   \frac{B_{2m}}{2m} \mid = \infty


    Exploit the fact that the zeta function:

    \zeta(2m) < 1

    and   (2m)   !  > \frac{(2m)^{2m}}{e^{2m}}

    I have to use the formula relating the zeta function to B_{2m} to prove this, but i cannot show that it is equal to infinity
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by ulysses123 View Post
    Prove that:


    lim_m _\rightarrow _\infty        \mid   \frac{B_{2m}}{2m} \mid = \infty


    Exploit the fact that the zeta function:

    \zeta(2m) < 1

    and   (2m)   !  > \frac{(2m)^{2m}}{e^{2m}}

    I have to use the formula relating the zeta function to B_{2m} to prove this, but i cannot show that it is equal to infinity
     B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}}\zeta(2n) . Applying Stirling's formula we get  |B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \right)^{2n} .
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  3. #3
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    i cant actually view this because it says to view images i need to have more than 10 posts, i am not sure why i cant see the solution
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by ulysses123 View Post
    i cant actually view this because it says to view images i need to have more than 10 posts, i am not sure why i cant see the solution
    I'm pretty sure that's just my signature you can't read... What can you see?
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  5. #5
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    [IMG]file:///C:/DOCUME%7E1/Apple/LOCALS%7E1/Temp/moz-screenshot.png[/IMG][IMG]file:///C:/DOCUME%7E1/Apple/LOCALS%7E1/Temp/moz-screenshot-1.png[/IMG]only the first bit where there is the equation for B(2n)
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  6. #6
    MHF Contributor chiph588@'s Avatar
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    Yeah, I gave you the equation for  B_{2n} and then said what it's asymptotic too. That's enough to show what you want.
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