An infinite series that arises in a numerical methods routine is _ n=2 ^ n^ 3 +5 n^ 4 -n
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
An infinite series that arises in a numerical methods routine is
n=2∑∞n4−nn3+5.
Answer the following questions about this series.
(1) Verify that the terms an=n4−nn3+5 are positive and eventually decrease for n≥2. Provide a clear argument that involves appropriate inequalities or derivative analysis.
(2) Compute the limit
n→∞limn1n4−nn3+5
and state its numerical value.
(3) Using the result of part (b) and an appropriate comparison test, determine whether the given series converges or diverges. Justify each step of your reasoning.