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

1. Feb 12, 2015

### japplepie

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?

2. Feb 12, 2015

### BiGyElLoWhAt

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

3. Feb 12, 2015

### BiGyElLoWhAt

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.

4. Feb 12, 2015

### japplepie

does this approximate derivative approach the actual derivative (as x grows to inf) ?

5. Feb 12, 2015

### BiGyElLoWhAt

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

6. Feb 12, 2015

### BiGyElLoWhAt

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.

7. Feb 12, 2015

### japplepie

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

8. Feb 12, 2015

### BiGyElLoWhAt

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.

9. Feb 12, 2015

### japplepie

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

10. Feb 12, 2015

### BiGyElLoWhAt

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

11. Feb 12, 2015

### BiGyElLoWhAt

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.

12. Feb 13, 2015

### japplepie

13. Feb 13, 2015

### BiGyElLoWhAt

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