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I have this question:

Consider the series

$e^{\tan(x)} = 1 + x + \dfrac{x^{2}}{2!} + \dfrac{3x^{3}}{3!} + \dfrac{9x^{4}}{4!} + \ldots $

Retaining three terms in the series, estimate the remaining series using "Little-$o$" notation with the best integer value possible, as $x\to 0$.

My question is:

What do they mean with "with the best integer value possible"? Someone who can point out the connection with little $o$ notation and a best integer vaulue possible?

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    Three terms in the series make $1+x+\frac12x^2$ hence the question asks for the smallest possible $n$ such that $\exp(\tan(x))=1+x+\frac12x^2+o(x^n)$ when $x\to0$.2011-09-08

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‘Best’ here would be ‘smallest’, so as Henning said, in all likelihood you’re being asked to write $e^{\tan x} = 1 + x + \frac{x^2}{2} + o(x^n)$ as $x\to 0$ for the smallest integer $n$ that makes the statement true.

Added: As Brugerfugl points out, that should be ‘largest’, since were’ looking at $x\to 0$.

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    @Brugerfugl: You’re quite right: I wasn’t thinking when I wrote that. My apologies; I’ll edit it.2011-09-10