Example of a sequence needed that satisfies certain conditions

Question

Can you please provide examples of sequence $\{a_1,a_2,\cdots\}$ such that $\sum_{i=1}^{\infty}a_i\to \infty$ while $\sum_{i=1}^{\infty}(a_i)^2<\infty$ with each $a_i\in[0,1)$. Thank you.

Answer 1

A classical example is the sequence $$ a = (1, 1/2, 1/3,1/4...) .$$ It is a well known result that $$ \sum_{k=1}^{\infty} a_k = \sum_{k=1}^{\infty} 1/k = \infty. $$ On the other hand, it holds that $$ \sum_{k=1}^{\infty} a_k^2 = \sum_{k=1}^{\infty} 1/k² = \pi²/6 \quad (\dagger). $$ Showing the equality in $(\dagger)$, hoewever, is nontrivial and is known as the Basler problem.

If you want the members of the sequence to be in $(0,1)$, then you can let it start at $a_2$ by defining $$ b = (1/2,1/3,...). $$ Obviously, $b$ is now another sequence fulfilling your requirements.

Answer 2

An interesting one is a_n = $\frac{1} p_{n}$, where p_n is the n'th prime. Then $\sum_{}^{} \frac{1} p_{n}$ diverges and $\sum_{}^{} \frac{1}p_{n}^2 $ converges by comparison to the reciprocals of squares.