Show that ∑i=0 to k nCi (-x)i is positive for all 0 ≤ x < 1/n and all k ≤ n, where nCi is the binomial coefficient.
Solution
The sum is (1 - nx) + (nC1 - nC2 x)x2 + (nC3 - nC4 x)x4 + ... . There may be an unpaired term at the end, but if so it is non-negative. A typical bracket is nCi(1 - x(n-i)/(i+1) ) > nCi (1 - 1/(i+1) ) > 0. So the sum is positive.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03