MathDB

Let

f: [0, 1] \to \mathbb{R}

be a continuous strictly increasing function such that

\lim_{x \to 0^+} \frac{f(x)}{x}=1.

(a) Prove that the sequence

(x_n)_{n \ge 1}

defined by

x_n=f \left(\frac{1}{1} \right)+f \left(\frac{1}{2} \right)+\cdots+f \left(\frac{1}{n} \right)-\int_1^n f \left(\frac{1}{x} \right) \mathrm dx

is convergent. (b) Find the limit of the sequence

(y_n)_{n \ge 1}

defined by

y_n=f \left(\frac{1}{n+1} \right)+f \left(\frac{1}{n+2} \right)+\cdots+f \left(\frac{1}{2021n} \right).

Problem Statement