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.