Convergence of this sum
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 :)
Interesting! The best I could come up with required no original thought and came from OEIS, sequence A073009:
Originally Posted by Bacterius
"This is also equal to ."
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 !
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. )
Not really that different from the earlier bound (their ratio goes to and becomes absurdly large)
Originally Posted by simplependulum
Thanks for all the help :)
I appreciate it.
I'll have to investigate this relationship with the integral more deeply. There might be something to see :o
This is called the "Sophmore's Dream". Only in a dream can (or some variation of that) always be true (Giggle).
Originally Posted by Drexel28
You may ask yourself what's the "Freshman's Dream"?. Well it's the fallacy , a common mistake amongst some highschoolers.
Interesting question, can you classify all (or some variant) such that ?
Originally Posted by chiph588@
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.(Doh)