A Limits of Taylor Series: Is $\sin x=x+O(x^2)$ Correct?

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The discussion centers on the expression of the Taylor series for sin x, specifically whether it is correct to write sin x = x + O(x^2). It is clarified that while this notation is technically acceptable, it implies a loss of information regarding the absence of the x^2 term. The correct representation is sin x = x + O(x^3), which indicates that the limit of (sin x - x)/x^3 is bounded as x approaches zero. Using O(x^3) effectively communicates that there are no terms of order x^2. Ultimately, the choice of notation impacts the clarity of the mathematical expression.
LagrangeEuler
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We sometimes write that
\sin x=x+O(x^3)
that is correct if
\lim_{x \to 0}\frac{\sin x-x}{x^3}
is bounded. However is it fine that to write
\sin x=x+O(x^2)?
 
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Yes, but you give away information. You could even say ##\sin x = O(x).##
 
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Writing ##O(x^3)## says there is no ##x^2## term.
 
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