Those are called (appropriately) limits, and they're the first thing you learn in Calculus.
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?
Paul's Site covers all of the basics.
If you look at the original post, he's really talking about sequences, not series.
There's not a whole lot of difference between taking the limit of a sequence and a function. Since you know limits for functions, you should be able to evaluate in the cases , , and . When x is negative, there are some difficulties with the definition of , but since you know n is an integer, it just breaks up into the two cases n odd or n even.
For the Fibonacci sequence, you can prove the formula by just checking the first two terms and the recurrence relation. Then you can see that .
also I dont know how you were able to determine that the intervals that needed to be checked are negative infinity to 0, 0 to 1, 1, and 1 to infinity. are those critical points or something?
also i did not know that there was a difference between series and sequence. i looked it up so now i know. in my OP I guess I am talking about a sequence, so how do I edit my OP? Is there a way I can change the thread title as well?
as for the fibonacci series, I'll get to that later.. when I understand more about what I am asking in my OP
Treat as a constant and look at the function as a function of just for a particular . If you think about it, it will make sense why you should consider in those particular intervals. It is not critical numbers. It is just basics of how numbers work. Numbers greater than one increase when you exponentiate them. Zero stays at zero. One stays at one. Numbers between zero and one get smaller. Then, just as hollywood said, negative numbers essentially do the same thing, as their absolute value, just alternating sign.