I am not able to solve this exercise. I've tried everything.

Exercise Let be $a_n > 0$ a sequence and suppose that $\sum a_n$ converges. Prove that exists a sequence $c_n > 0$ such that $\lim c_n = \infty$ and $\sum c_na_n < \infty$

I don't even know if I have to find an "always-valid" closed form for $c_n$ (maybe in function of $a_n$?) or prove its existence in another way.

Thanks in advance.


Hint: There exists a sequence of positive integers $n_1< n_2 < \cdots$ such that $\sum_{n=n_k}^{n_{k+1}-1} a_n < 1/2^k.$ Recall that $\sum_k k/2^k < \infty.$

  • $\begingroup$ Thank you, but I cannot understand how to use this $\endgroup$ – Marco Nervo May 7 '17 at 18:38
  • $\begingroup$ @MarcoAll-inNervo I edited it a bit. Maybe this is more helpful $\endgroup$ – zhw. May 7 '17 at 18:46
  • $\begingroup$ Thank you very much, this really helps! Thanks especially for changing $\infty$ with $n_{k+1} - 1$ $\endgroup$ – Marco Nervo May 7 '17 at 18:56

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