Example of sequence such that $a_n\to0$ but $\sum a_n^2$ diverges

Question

I'm looking for a simple sequence $a_n = (a_1,a_2,a_3,...)$ such that $\lim_{n\rightarrow \infty}a_n = 0$ but the sum $(a_1)^2+(a_2)^2+(a_3)^2+...$ does not converge. I was thinking we could use the geometric sum formula $1/(1-a)$ such that $a=1$ (for example, a sequence such as $(2,1,1/2,1/4,...)$, but this doesn't seem to work.

Answer 1

Hint: consider the sequence $a_n=\frac{1}{\sqrt{n}}$

Answer 2

HInts: $a_n = \frac{1}{\ln n}$ is also okay.

Answer 3

There are good answers for your question but I like to explain why they satisfy your given conditions.

First, note that $$\lim_{n\to\infty}a_n=0⟺\lim_{n\to\infty}a_n^2=0.$$ Therefore if we replace your $a_n$ by $b_n=a_n^2 \,\,\,\forall n\in\mathbb{N},$ then

We have to find a sequence of positive real numbers $(b_n)$ such that $\lim_{n\to\infty}b_n=0$ and $\sum_{n=1}^{\infty}b_n$ does not convergent.

Additionaly,
If $\sum_{n=1}^{\infty}b_n$ convergent, then $\lim_{n\to\infty}b_n=0.$ Here the converse may not true.