I'm studying convergence and divergence of infinite sums. I've been working on this but I can't figure it out applying the ratio criterion and the root one. The exercise asks to determine whether it diverges or not. This is the sum:
$$\sum_{n=1}^\infty \frac{n(n-1)}{(n+1)(n+2)(n+3)}$$ Ii've supposed it diverges since it's similar to $\sum_{n=1}^\infty \frac{1}{n}$ Thank you in advance.
Yes your guess is right, to prove that rigorously let apply limit comparison test that is
$$\frac{\frac{(n)(n-1)}{(n+1)(n+2)(n+3)}}{\frac1n}=\frac{(n^2)(n-1)}{(n+1)(n+2)(n+3)}\to 1$$
to conclude that the series diverges.
As an alternative, by direct comparison test we can use that as $n>1$
$$\frac{(n)(n-1)}{(n+1)(n+2)(n+3)} \ge \frac{(n-1)^2}{(n+3)^3}=\left(\frac{n-1}{n+3}\right)^2\frac1{n+3}\ge \frac1{25} \frac1{n+3}$$
$$\frac{n(n-1)}{(n+1)(n+2)(n+3)}$$ divide numerator and denominator by $n^3$
$$\frac{\frac{n}{n}\cdot \left(1-\frac{1}{n}\right)\cdot \frac{1}{n}}{\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\left(1+\frac{3}{n}\right)}$$ as $n\to\infty$ it is equivalent to $\frac{1}{n}$ which diverges.
