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The statement about sine above is not something mathematicians would write. It makes little sense to use big-O notation in this context, as it doesn't say anyth
by xyzzyz 9y ago
The statement about sine above is not something mathematicians would write. It makes little sense to use big-O notation in this context, as it doesn't say anything useful here: the O(x^7) element absolutely dominates the remaining explicit elements of lower order, so including them tells us absolutely nothing. In fact, sin(x) = O(1).
However, mathematicians do indeed use similar notation in this context, that is, little-o notation. It is in fact true that
sin(x) = x -x^3/6 + x^5/120 + o(x^5), x -> 0.
- gizmo686 9y agohttps://en.wikipedia.org/wiki/Taylor_series#First_example https://en.wikipedia.org/wiki/Taylor_series#First_example https://www.wolframalpha.com/input/?i=taylor+series+sin+x https://www.wolframalpha.com/input/?i=taylor+series+sin+x Notice that in your example, you have o(x^5) and an explicit x^5 term. In my example I have O(x^7), but no explicit x^7 term. It is true that I cannot think of a circumstance where you want to do this abuse of notation and would care if you were forced to use little-o or big-O instead of the other. In my experience, it happens to be more common to use big-O.
- j2kun 9y agoI have definitely used big-O as the parent described. I think many mathematicians would write it in that way.
- proto-n 9y agoHave you considered that higher orders of x are in fact smaller when x is near 0? The parent comment was right and you are wrong, around zero x^5 absolutely dominates x^7 and the big-O notation is used. See for example here [1] [1] https://en.wikipedia.org/wiki/Taylor_series#First_example https://en.wikipedia.org/wiki/Taylor_series#First_example