MathDB

11

Part of 1996 IMC

Problems(1)

IMC 1996 Problem 11

Source: IMC 1996

3/4/2021
i) Prove that lim⁡x→∞ ∑n=1∞nx(n2+x)2=12 \lim_{x\to \infty}\,\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}=\frac{1}{2}. ii) Prove that there is a positive constant cc such that for every x∈[1,∞)x\in [1,\infty) we have ∣∑n=1∞nx(n2+x)2−12∣≤cx\left|\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}-\frac{1}{2} \right| \leq \frac{c}{x}
limitreal analysis