High School Why is the %change of the %change of a sequence so chaotic?

[beamthegreat]

Take the sequence 1,2,3,4,5,6,7,8,9,10...
If you found the percent change for each interval and kept on finding the percent change of the percent change of the sequence, why does the change become more and more chaotic?

Here is a quick table I made:
https://docs.google.com/spreadsheets/d/1gufw9MEJBUz_YEZBcfu6F_9IPAS3umoDyYh-4aXQRUA/pubhtml

Are there any explanation for this? There doesn't appear to be any obvious pattern to it.

 

[andrewkirk]

It's just the propagation of a tiny rounding error in the spreadsheet calculation, that grows with each step to the right. At first it is invisible within the digits shown but it grows to become visible and ultimately to dominate.

In fact the correct number in row j of column k is $-1/(j-1)$, for $k>1$ and $1/(j-1)$ for $k=1$. So the correct numbers are the same across every row, other than the first column, and diminish methodically in absolute value as we go down a column. It's fairly straightforward to prove this algebraically.

 

[Simon Bridge]

for the sequence n = 1,2,3,4... verify the calculation analitically:
the relative change $c_n$ in going from n-1 to n is $c_n = [n-(n-1)]/n = 1/n$
(note: the percentage change is $100c_n$)

Now you have a sequence of relative changes: $c_1, c_2 \cdots$
where $c_n = 1/n$

the relative change going from $c_{n-1}$ to $c_n$ is:
$$d_n = \frac{c_n-c_{n-1}}{c_n}=\cdots$$... complete the expression.

 

[mfb]

In this case, it is just rounding errors, but you'll also see this without rounding errors if the original sequence is a bit chaotic. The reason is the nonlinearity: if one difference in a sequence happens to be very small, it appears in the denominator in one element in the next sequence, producing a huge value. Change the 6 to 6.1 for example and see how the values change dramatically even in the range where they are nice with the current sequence.