Mathematics

Problem

Consider the series

\displaystyle\sum_{n=1}^{\infty}\frac{n}{n^{3}+1}

. Which of the following comparisons is sufficient to conclude that this series converges? a) Because for all

n\ge 1,\; \dfrac{n}{n^{3}+1}\ge \dfrac{1}{n^{3}}\;

and

\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^{3}}

converges. b) Because for all

n\ge 1,\; 0\le\dfrac{n}{n^{3}+1}\le\dfrac{1}{n^{2}}\;

and

\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^{2}}

converges. c) Because for all

n\ge 1,\; 0\le\dfrac{n}{n^{3}+1}\le\dfrac{1}{n}\;

and

\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n}

diverges. d) Because for all

n\ge 1,\; 0\le\dfrac{n}{n^{3}+1}\le\dfrac{1}{\sqrt{n}}\;

and

\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{\sqrt{n}}

diverges.