Take the 2-minute tour ×
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.

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|improve this question
add comment

1 Answer 1

  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|improve this answer
add comment

Your Answer

 
discard

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.