Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given $f$, a uniformly continuous function defined on the interval $[0,1]$, I need to prove that $$\lim_{n\rightarrow \infty} \frac{1}{2^n} \sum_{k=1}^n (-1)^k \binom{n}{k} f(k/n)=0.$$ I have tried tackling this exercise from a couple of angles but I seem to lack the intuition and technical skills to crack this egg open, so I am at your mercy.

share|cite|improve this question
  1. As $f$ can be expressed as a uniform limit of polynomials, by linearity it's enough to show the result when $f(x)=x^p$ where $p$ is a non-negative integer.

  2. We can show this by induction on $p$, using the relationship $$\frac{k^p}{n^p}\binom nk=\frac{k^{p-1}}{n^{p-1}}\binom{n-1}{k-1}.$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.