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There is given positive decreasing sequence of real numbers $a_{1},a_{2},a_{3},...$

  • Calculate the sum in nice form: $$SUM=\sum_{i=1}^{N}a_{i}-2\sum_{1\le i<j\le N}a_{i}a_{j}$$
  • If it is impossible to do it in general then let try it in case when $a_{n}=\frac{(-1)^{n}}{n^\sigma}$ where $0<\sigma<1$.
  • Find lower bound of the $SUM$

After few transformations i got $SUM=(\sum_{n=1}^{N}(-1)^{n}a_{n})^{2}-4\sum_{n=1}^{N}a_{n}\sum_{m=1}^{N-n}a_{n+2m}$

But is this going to help me?

Second idea is about considering these elements of sequence as a variables of polynomial(which degree is equal to $2$)

Any hint?

Regards.

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  • $\begingroup$ Huh? $a_n=(-1)^n/n^\sigma$ does not satisfy $(a_n)$ decreasing. $\endgroup$ – user10354138 May 17 at 17:17
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    $\begingroup$ Hint: $2\sum_{i<j} a_i a_j = \sum_{i\ne j}a_i a_j = \left(\sum_i a_i\right)^2 - \sum_i a_i^2$ If you sum stop at $\infty$ instead of finite $N$, the sum you have can be expressed in terms of zeta functions. $\endgroup$ – achille hui May 17 at 17:22

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