# How to approximate the rate of growth of an integer sequence?

## Main Question or Discussion Point

Assume that I have absolutely no clue to what is the formula used to generate a sequence.

How do I know what kind of formula that is? (Exponential / Linear / Polynomial / etc)

Also assume that there is only 1 formula that generates the sequence.

I have read somewhere before that:

f'(x) ~ f(x+1)-f(x) as x -> infinity; is this true?

BiGyElLoWhAt
Gold Member
change in y over change in x (which is 1, because it takes integer arguments) is the slope, sooo...

BiGyElLoWhAt
Gold Member
Actually, I neglected that as x goes to infinity part. I'm not so sure about that. Seems sketchy. That will give you the approximate derivative at some value x.

Actually, I neglected that as x goes to infinity part. I'm not so sure about that. Seems sketchy. That will give you the approximate derivative at some value x.
does this approximate derivative approach the actual derivative (as x grows to inf) ?

BiGyElLoWhAt
Gold Member
Hmm... I'm trying to think about it, and analytically, it seems to work, at least for polynomial expressions.

BiGyElLoWhAt
Gold Member
But, if you had a sequence f such that $f:=cos(n\pi)$ this would not work for retrieving the derivative of a continuous function.

I just checked in wolfram, it also does not work for f(x)=x^x :c

BiGyElLoWhAt
Gold Member
Yea, I don't know enough about derivatives of piecewise functions, because that's what a sequence is, to be able to solve something so generally. Sometimes it works, sometimes it doesn't.

Yea, I don't know enough about derivatives of piecewise functions, because that's what a sequence is, to be able to solve something so generally. Sometimes it works, sometimes it doesn't.
But could the fact that I can create an infinitely more terms make it more precise?

Btw, this is the one of the sequences that I'm currently working on.. Just by inspecting it, I could tell that its close to 4^x

1
30
185
886
3855
16064
65569
264930
1065059
4270948
17105253
68463974
273941863
1095939432
4384101737
17537095018

BiGyElLoWhAt
Gold Member
I guess you could check that by doing 4^11-4^10 and checking it with d(4^x)/dx at x=10 then doing the same with 4^12-4^11 and checking it with d(4^x)/dx at x=11

BiGyElLoWhAt
Gold Member
The thing of it is, even if it did converge to at infinity, that definition would presumably be less than useful at values of x less than infinity. I'm not trying to contradict what you guys are doing in class, I'm just throwing out my honest opinion. Perhaps this is a good point for someone else to chime in here.

The thing of it is, even if it did converge to at infinity, that definition would presumably be less than useful at values of x less than infinity. I'm not trying to contradict what you guys are doing in class, I'm just throwing out my honest opinion. Perhaps this is a good point for someone else to chime in here.[/QUOTE}
It's ok, I'm not looking for the 100% precise function that makes these sequences, I just want to have an approximate rate of growth and it is asymptotic to this approximation as x -> infnity.

BiGyElLoWhAt
Gold Member
True, it does seem to approach f'(x) as x approaches infinity, for x's close to infinity.