Sum of sequence which is not geometric or arithmetic

[Elearner]

If you have a sequence which is not geometric or arithmetic, is there any methodology to follow in order to have a formula for its sum ?

Take for example the following sequence: \(\displaystyle \{0.9^{\frac{1}{2}(n-i+1)(i+n)}\}_{i=1}^n\). It is not a geometric or an arithmetic progression. I don't see how to split it into sums of sequences which are arithmetic or geometric. Is there any hints I can get to proceed with writing a formula for this sum ?

\(\displaystyle S_n = \sum_{i=1}^n 0.9^{\frac{1}{2}(n-i+1)(i+n)} \)

 

[tkhunny]

I tried to fix your math code. We use [ t e x ] followed by [ / t e x ] around here. I'm not perfectly confident that I have reproduced your desired result. Have you ANY examples of a more convenient sum being found for anything?

 

[Elearner]

I tried to fix your math code. We use [ t e x ] followed by [ / t e x ] around here. I'm not perfectly confident that I have reproduced your desired result. Have you ANY examples of a more convenient sum being found for anything?

Thanks tkhunny. It seems that there is no closed-form expression for this sum. Please find more details on my question here:
https://math.stackexchange.com/questions/2433266/sum-of-a-sequence-which-is-neither-arithmetic-nor-geometric

I am now trying to figure out if there is any way to prove that \(\displaystyle \forall n'>n>5: S_{n'} \leq S_n\) ?

 

[Elearner]

Any thoughts/ideas about my last question ?