MathDB

Problem 3

Part of 2001 VJIMC

Problems(2)

ln2*ln3*...ln n < sqrt(n!)/n

Source: VJIMC 2001 1.3

7/26/2021
Let n≥2n\ge2 be a natural number. Prove that ∏k=2nln⁡k<n!n.\prod_{k=2}^n\ln k<\frac{\sqrt{n!}}n.
inequalities
lim xf(x)=0 if f:R+-&gt;R+ and int^&infin;_0&lt;&infin;

Source: VJIMC 2001 2.3

7/21/2021
Let f:(0,+∞)→(0,+∞)f:(0,+\infty)\to(0,+\infty) be a decreasing function which satisfies ∫0∞f(x)dx<+∞\int^\infty_0f(x)\text dx<+\infty. Prove that lim⁡x→+∞xf(x)=0\lim_{x\to+\infty}xf(x)=0.
calculuslimitsintegrationfunction