MathDB
Find the limit in terms of A and B

Source: 2019 Jozsef Wildt International Math Competition-W. 21

May 18, 2020
integrationlimit

Problem Statement

Let ff be a continuously differentiable function on [0,1][0, 1] and m∈Nm \in \mathbb{N}. Let A=f(1)A = f(1) and let B=∫01x−1mf(x)dxB=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx. Calculate lim⁡n→∞n(∫01f(x)dx−∑k=1n(kmnm−(k−1)mnm)f((k−1)mnm))\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{n^m}\right)\right)in terms of AA and BB.
Back to ProblemsView on AoPS