Let for . Respond to each part thoroughly, showing all relevant algebraic and…

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

Problem

Let cn=ln⁡nn3c_n = \dfrac{\ln n}{n^{3}} for n≥2n \ge 2. Respond to each part thoroughly, showing all relevant algebraic and analytical work. (a) Evaluate the improper integral K=∫2∞ln⁡xx3 dx\displaystyle K=\int_{2}^{\infty} \frac{\ln x}{x^{3}}\,dx by an appropriate method of your choice. Present each step of your calculation and state any integration techniques you employ. (b) Decide whether the series ∑n=2∞cn\displaystyle \sum_{n=2}^{\infty} c_n converges or diverges. Justify your decision using the result from part (a). (c) For a fixed integer N≥2N\ge 2, express an inequality that bounds the remainder RN=∑n=N+1∞cnR_N = \displaystyle \sum_{n=N+1}^{\infty} c_n in terms of an improper integral, and write the resulting bound explicitly.

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