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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.

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up vote 2 down vote accepted

Try $b_k=0$ for every $k$, $a_k=2^{-n}$ if $k=4^n$, and $a_k=0$ otherwise.

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Oh!That's great!I never think in this way!Thank you! – Juntao Huang Jun 4 '12 at 16:12

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