# Thread: Inequality with binomial coefficients

1. ## Inequality with binomial coefficients

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

2. ## Re: Inequality with binomial coefficients

Put $\displaystyle f_n(x):=\sum_{i=0}^n(-1)^i\binom nix^{2+2i}$ and compute $\displaystyle \int_0^1f_n(t)dt$.

3. ## Re: Inequality with binomial coefficients

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