Divergence Test Question

Question

How would I show this series diverges

$$\sum_{r=1}^{\infty} \frac{(-1)^rr^3}{2r^3+3r^2+1}$$

It's a monotonically increasing sequence, so i know the series would diverge, but how would i prove this?

Answer 1

This is an application of three useful results in basic analysis.

Fact 1. Let $\{a_n\}$ be a sequence. Then $a_n\to 0$ if and only if $|a_n|\to 0$.

Fact 2. Let $\{a_n\}$ be a sequence that does not converge to $0$. Then $\sum a_n$ diverges.

Fact 3. Let \begin{align*} p(t) &= a_m t^m+a_{m-1}t^{m-1}+\dotsb+a_1t+a_0 \\ q(t) &= b_n t^n+b_{n-1}t^{n-1}+\dotsb+b_1t+b_0 \end{align*} where $a_m\neq0$ and $b_n\neq0$. Then $$ \lim_{t\to\infty}\frac{p(t)}{q(t)}= \begin{cases} 0 & m<n \\ a_m/b_n & m=n \\ \lambda \infty & m>n \end{cases} $$ where $\displaystyle\lambda=\frac{a_m}{|a_m|}=\operatorname{sign}(a_m)$.

Can you put these facts together to see why your series diverges?