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Math Help - Inequality with binomial coefficients

  1. #1
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    Inequality with binomial coefficients

    I am trying to prove for n = 0, 1, 2, ...
    \sum_{i=0}^n \frac{(-1)^i}{3+2i}\binom{n}{i}>0
    The attached Matlab code in ineq.txt produces a plot that seems to justify the inequality (Inequality with binomial coefficients-ineq.jpg).
    I know from the Binomial Theorem
    \sum_{i=0}^n (-1)^i\binom{n}{i}=0
    Further, the scaling term 1/(3+2i) is crucial in the first inequality. For instance, the scaling term 1/(3+2i^2) would violate the inequality (Inequality with binomial coefficients-ineq2.jpg). Could anyone help me with a formal proof of the inequality? Thanks!
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  2. #2
    Super Member girdav's Avatar
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    Re: Inequality with binomial coefficients

    Put f_n(x):=\sum_{i=0}^n(-1)^i\binom nix^{2+2i} and compute \int_0^1f_n(t)dt.
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  3. #3
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    Re: Inequality with binomial coefficients

    Indeed,
    f_n(x):=\sum_{i=0}^n(-1)^i\binom nix^{2+2i}=x^2(1-x^2)^n>0
    for x\in(0,1), such that
    0<\int_0^1f_n(t)dt=\sum_{i=0}^n \frac{(-1)^i}{3+2i}\binom{n}{i}
    Many thanks for your quick reply!
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