Consider the series V= _ n=1 ^ n^ 3 +2 n^ 7 +5

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

Problem

Consider the series

V=\sum_{n=1}^{\infty}\frac{n^{3}+2}{\sqrt{n^{7}+5}}.

For

n\ge 1

denote

v_n=\dfrac{n^{3}+2}{\sqrt{n^{7}+5}}

. (1) Show through dominant-term analysis that for large

n

v_n\sim \frac{1}{n^{1/2}}.

Present each algebraic step leading to this simplification. (2) Compute the limit

L=\lim_{n\to\infty}\frac{v_n}{\dfrac{1}{n^{1/2}}}

and verify that the limit is finite and non-zero. (3) Using the limit from part (b) and the known behavior of

\sum \dfrac{1}{n^{1/2}}

, decide whether the series

V

converges or diverges. Provide complete reasoning.