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What is the minimal value of the sum $$a_1^{-2}+\sum_{k=2}^\infty 2(2^k-2)2^{-2k} a_k^{-2}$$ given $a_1+a_2+...=1$ and all $a_j$'s are positive?

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  • $\begingroup$ What have you tried? Do you think this is a tricky problem in some way, with a counterintuitive answer? $\endgroup$ Sep 6, 2017 at 15:33
  • $\begingroup$ I have found a sequence with the value less than 130. I guess that the answer does not have an explicit form, or has a very complicated explicit form. So the best way may be to merely calculate the answer. But I do not know how. $\endgroup$
    – Viktor
    Sep 6, 2017 at 15:39
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    $\begingroup$ Have you tried the method of Lagrange multipliers? Write your expression as $S=\sum_{k\ge1} c_k/a_k^2$, differentiate $S+\lambda(\sum_{k\ge1}a_k-1)$with respect to the $a_k$, and so on... $\endgroup$ Sep 6, 2017 at 16:07
  • $\begingroup$ Thank you. It seems that Lagrange multipliers yield a unique solution. $\endgroup$
    – Viktor
    Sep 6, 2017 at 16:19

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