I have observed that in these two series':
1. (x^2), (x^4), (x^6)....
2. (x^3), (x^5), (x^7)....
it becomes more and more rigid and sharp. By mere inspection of the graphs I guess we can say that for the first series it is a function that is a parabola but as we go forward infinitely in the series it approaches (probably asymptotically) something like the shape below:
same thing with the second series, towards infinity it approaches a shape that looks like:
And recently I read that some guy(centuries ago) was able to prove that as you extend the fibonacci series into infinity it approaches asymptotically the value or shape of the golden ratio. So in the same way I guess we can also prove mathematically the observations that I have made above based on inspection.
So does anyone know what is the method of calculating what is the limit as infinity is being approached of an entire series?