text: # Calculate the sum.
  
  There is given positive decreasing sequence of real numbers $$a_{1},a_{2},a_{3},...$$
  
  • Calculate the sum in nice form: $$SUM=\sum_{i=1}^{N}a_{i}-2\sum_{1\le i
  • If it is impossible to do it in general then let try it in case when $$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.
  
  • Huh? $a_n=(-1)^n/n^\sigma$ does not satisfy $(a_n)$ decreasing. – user10354138 May 17 at 17:17
  • 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. – achille hui May 17 at 17:22
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