Consider the series where

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the series ∑n=2∞an\displaystyle\sum_{n=2}^{\infty}a_n where an=ln⁡nn2a_n=\dfrac{\ln n}{n^{2}}. Define f(x)=ln⁡xx2f(x)=\dfrac{\ln x}{x^{2}} for x≥2x\ge 2. (1) Show that ff is positive, continuous, and decreasing on the interval [2,∞)[2,\infty). Provide algebraic or calculus–based reasoning for each property. (2) Explain how the properties established in part (a) affect the convergence or divergence of the series ∑n=2∞an\displaystyle\sum_{n=2}^{\infty}a_n. State your conclusion about the series and justify it thoroughly.

Related theory

Browse the problem library