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I am reading this document.

In this article after defining strong derivative Knuth goes on to calculate derivative of $x^n$. There he uses definition of strong derivative to expand $(x+\epsilon)^{n+1}$ as follows,

$(x+\epsilon)^{n+1} = (x+\epsilon)[x^n + d_n(x)\epsilon + \mathcal{O}(\epsilon^2)]$

when I expand the right side I get

$x^{n+1} + (x d_n(x) + \ x^n)\epsilon + \color{red}{x \ \mathcal{O}(\epsilon^2) + \epsilon^2 \ d_n(x) + \epsilon \ \mathcal{O}(\epsilon^2)}$

But in Knuths calculations the red part is just $\mathcal{O}(\epsilon^2)$.

Question is how? I don't know how to work this out in detail.

  • 0
    This is explained [here](http://en.wikipedia.org/wiki/Big_O_notation#Properties). Also, [this question](http://math.stackexchange.com/questions/25262/the-big-o-notation) is similar, and it answers your question.2012-02-20

1 Answers 1