Is $\frac{1}{n^2}\sum_{j=n}^{\infty} \frac{1}{j}$ bounded?

Question

Repeating the title: is $\frac{1}{n^2}\sum_{j=n}^{\infty} \frac{1}{j}$ bounded?

Note that the sum starts from $n$.

(I am guessing it must not be, otherwise I will have a puzzle on a theorem I am trying to use. Nevertheless, I would like to understand this. It has been a thousand years since I dealt with convergences of series and I'm pretty rusty.)

Answer 1

If you really want to define such quantity, then you have that $\frac1{n^2}\sum_{j=n}^\infty\frac1j=\infty$ for all $n$ because $\sum_{j=n}^\infty\frac1j$ diverges to $\infty$. Therefore the sequence would be constant $=\infty$.