Comment #9 on issue 3135 by [email protected]: Multiple series expansions
http://code.google.com/p/sympy/issues/detail?id=3135
The O that you know from computer science is order at infinity. The O used
in series is order at 0. You can think of the two as basically opposite.
For order at infinity, larger exponents dominate, because they grow faster
at infinity.
For order at 0, smaller exponents dominate. This is because if x is near 0,
x**n is smaller for larger n.
Or, if you prefer the limit ratio definition of O, you can think like this.
For order at infinity, x is O(x**2) because x/x**2 -> 0 as x -> oo. For
order at 0, we do the same thing except we take x -> 0. In that case x/x**2
-> oo as x -> 0. The reverse is true, x**2/x -> 0 as x -> 0, so x**2 is
O(x) at x = 0.
The idea of "dominance" is important for series expansion because the most
important terms are the dominant ones. So if your series expansion is 1 + x
+ x**2 + ... near x = 0, then the smaller degree terms will be more
important. Try plugging 0.01 into 1, 1 + x, 1 + x + x**2, and so on. Each
term you add makes less and less of a contribution. We can just say the
remainder of the terms are O(x**3), meaning the contribution is on the
order of x**3 (this is closely related to the so-called Taylor error term).
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