Question

For what r does `3/(n^(2r - 3))` converge or diverge?

Answer 1

For `r>=3/2` and only for them.

Explanation:

Let `alpha` be a real number.
The sequence `(1/n^alpha)_(n \in NN^*)` converges if and only if `alpha >= 0`.

Of course, `(1/n^alpha)_(n \in NN^*)` converges iff `(3/n^alpha)_(n \in NN^*)` converges...

So, here, the sequence converges iff `2r-3>=0` iff `r>=3/2`.