I'm trying to follow some notes my supervisor has written and I've got the first three terms of a Taylor series $1 - (-k \lambda) + (-k\lambda)^2 /2$ becomes $k\lambda +O(k\lambda^2)$ Is this correct? What happened to the "1"?
$1 - (-k \lambda) + (k\lambda)^2 /2 = (k\lambda +O(k\lambda^2))$?
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taylor-expansion
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0There is a distinction between 'little-o' and 'big-o'. I am presuming you intended the latter? – 2012-05-29
1 Answers
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The Taylor polinomial of order $n$ of $e^{-x}$ is
$e^{-x}=1-x+\frac{x^2}{2!}-\cdots+\frac{(-1)^nx^{n}}{n!}+o(x^{n})$
Most probably, what he wrote is that
$e^{-({-\lambda k})}=1-({-\lambda k})+\frac{(-\lambda k)^2}{2!}+o((-\lambda k)^{2})$
$e^{{\lambda k}}=1+\lambda k+\frac{\lambda^2 k^2}{2!}+o((\lambda k)^{2})$
I don't see why $(a)$ he disregarded the $1$, $(b)$ he disregarded the term of degree $2$. Maybe he subtracted the series with that of $e^{-\lambda k}$?
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0In this situation the 'big-O' notation is also correct. It was my inference that was wrong. – 2012-05-29