Why does this series diverge? What's wrong with my reasoning?

Question

So I have this series

$$\sum_{j=2}^{\infty} \frac{-(2j-3)(j-3)}{(5j-8)(4j+1)}$$

I figured by reasoning that with the leading coefficients: -2J^2 / 20 j^2

the bottom would win out if the J went to infinity.

Also it is ratio of 1/10 so I thought the series would converge.

So why does this series diverge?

Can someone use the Cauchy Condensation, Comparison Test, or Ratio test.

I haven't started reviewing integral test or the other tests yet.

Answer 1

It diverges: for any convergent series, the general term tends to $0$. Here the general term tends to $-\dfrac1{10}$.

Answer 2

The expression $$ -\frac{(2j-3)(j-3)}{(5j-8)(4j+1)}=-\frac{(2-3/j)(1-3/j)}{(5-8/j)(4+1/j)}$$ has a limit of $-(2)(1)/(5)(4)=-1/10$ as $j\to \infty.$ Since the terms of the series do not tend to $0,$ the series has no limit.

Answer 3

I see the flaw in my reasoning. I have done the ratio test and cauchy condesation test and found that it diverges.