Mathematics

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}}

. a. 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. b. 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. c. 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.