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Math Help - proving inequality

  1. #1
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    proving inequality

    so i started studying calculus and stumbled upon an inequality I'm having difficulties to prove,

    for every 1<=k<=n prove
    C(n,k)*1/(n^k)<=1/(2^(k-1))
    so i brought it to
    C(n,k)*2^k<=2*n^k
    and now i tried induction and it got to newton's binomial, and then im stuck...

    looks something like

    C(n,k-1)*2^k+c(n,k)*2^k<=2*(1+n+n^2...kn^k-1+n^K) used c(n+1,k)=c(n,k-1)+c(n,k)
    (dont know how to write epsilon so its open ^^^^^ )
    while the red thingy is the assumption of the induction,
    how i prove the pink part??

    please, require assistance!
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  2. #2
    Super Member girdav's Avatar
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    Re: proving inequality

    Hint: write n^{-k}\binom nk=\prod_{j=0}^{k-1}\left(1-\frac kn\right)\cdot \frac 1{k!}\leqslant \frac 1{k!}.
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  3. #3
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    Re: proving inequality

    I don't understand, why?
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