I meet a problem in a textbook( Page 125 in Partial differential Equation written by Fritz John).
Find sequences $a_k$, $b_k$ for which the series $\sum\limits_{k=1}^{\infty}k(a_k^2+b_k^2)$ diverges, while the series $\sum\limits_{k=1}^\infty(\left|a_k\right|+\left|b_k\right|)$ converges.
I cannot find such sequences.