Find a real sequence $x_n$ such that $n x_n \to 0$ as $n \to \infty$ and $\sum_{n=1}^{\infty} x_n$ diverges

Question

I'm stuck on trying to find a real sequence $x_n$ such that $n x_n \to 0$ as $n \to \infty$ and $\sum_{n=1}^{\infty} x_n$ diverges. Here's what I have tried so far:

I've started with trying to find $x_n$ as a real-valued function of $n$. In order to have $n x_n \to 0$, we require $$\frac{x_n}{1/n} \to 0$$ and since the denominator tends to $0$ as $n$ increases, we require that $x_n \to 0$. We also require that $x_n$ tends to $0$ "faster" than $1/n$ does, in order for the above quotient to go to $0$.

However if you try functions such as $x_n=1/n^2$ or $x_n=n/2^n$ (for example) which tend to $0$ faster than $1/n$ does, then you see that $\sum_{n=1}^{\infty} x_n$ converges and I haven't been able to find a function for $x_n$ that satisfies the above conditions and also has a divergent infinite sum. It's certainly possible that the form for $x_n$ might not be a simple elementary function of $n$, but I don't know how you'd go about finding any "unusual constructions" that work.

Answer 1

Take $x_n = \dfrac 1 {n\ln n }.$