Examine the series U= _ n=1 ^ 3n+2 n^ 3 +1

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

Problem

Examine the series

U=\sum_{n=1}^{\infty}\dfrac{3n+2}{\sqrt{n^{3}+1}}.

Answer each part below. (1) Identify a simpler positive–term series

V

whose convergence or divergence can be decisively established and that can be used to compare with

U

. State

V

explicitly and justify why it is an appropriate choice. (2) Using an appropriate comparison, determine whether

U

converges or diverges. Show all steps in your comparison and state any limits you evaluate. (3) Classify

U

as absolutely convergent, conditionally convergent, or divergent. Explain your reasoning clearly.