"This is also equal to ."
In chemistry class, instead of working hard on valence shells and bonds, I've been fiddling with my calculator. I tried to evaluate the following sum out of interest :
It converges. Has anyone ever seen this actual sum somewhere ? Does it have a particular meaning or use, as ? Or is it just yet another random and useless summation ? I'm asking the question because I feel this sum has something special, having the variable both in base and exponent.
I've been looking on Google but didn't find any reading on it. Any input is appreciated
Although I am not a bookworm , all of the books i have read never mention how to find the sum in an analytical way, at most they only explain the relation with the integral . I think the most reasonable way to find the sum is using the calculator because the series converges so fast !
This raises the question of the remainder when the sum is truncated after terms:
I can show that , but I expect someone can do better (there is plenty of elbow room for improvement for largish ).
I am not sure ... please check it
I consider and let
be the sum of the area of the rectangles with dimensions ,
which is smaller than the curved area under since is decreasing .
If my calculations are all correct , it is interesting that the infinite sum starting from the next term is always smaller than that term . ( i.e. )
This series can be extended to a more general function:
You can check it and this function is define for all real and it's probably monotonically increasing (just found experimentally). It's non-negative for all real .
Besides, straightly from the Taylor expansion of the function:
But this is just trivial.