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I have a (perhaps silly) question, but I am reviewing Taylor series approximation and looking at this slide. In the third panel, $f(a+h)$ is isolated but I am confused why $O(h^{2})$ does not have a minus in front of it.

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  • $\begingroup$ Are you familiar with the big-O notation? $\endgroup$
    – MathNewbie
    May 22, 2015 at 3:21
  • $\begingroup$ No not really - I see they are defining this qty on the second slide. $\endgroup$
    – B_Miner
    May 22, 2015 at 3:31
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    $\begingroup$ I guess ... it uses O as a bundle of remainder hence it is not important using sign 'plus or minus' since O($h^2$) is much smaller than others $\endgroup$
    – user128766
    May 22, 2015 at 3:43
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    $\begingroup$ I can't read the question, but the big-O notation is often used sloppily. It typically means one of a class of functions that satisfies a given property (such as $\lim_{h\to 0} {r(h) \over h^2} = 0$). Hence (constant) scalar multiples are bandied amount without regard to arithmetic. $\endgroup$
    – copper.hat
    May 22, 2015 at 4:49
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    $\begingroup$ @copper.hat +1 for "big-O notation is often used sloppily." It often causes great confusion when it is used to mean "lesser than or equal to" rather than "equal to" (since big-O really only denotes an upper bound on the growth of the function) $\endgroup$
    – MT_
    May 22, 2015 at 4:59

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In this case, the instructor is using "$O(h^2)$" to say "plus some other miniscule term." Since this term is negligent, it doesn't matter whether there is a plus sign or minus sign is in front of it. In general, you will find that big-O notation is often used "sloppily" like this.

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