Definition of asymptotic relation

[hanson]

Hi all!

Can anyone explain to me why the asymptotic relation between a function and a power series is defined in such a way:

For all N,

<br /> f(x) - \sum_{n=0}^N a_n(x-x_0)^n &lt;&lt; (x-x_0)^N <br />

How does this incorporate the idea of asymptoticity?

Please kindly help.

 

[arildno]

Hi all!

Can anyone explain to me why the asymptotic relation between a function and a power series is defined in such a way:

For all N,

<br /> f(x) - \sum_{n=0}^N a_n(x-x_0)^n &lt;&lt; (x-x_{0})^{N} <br />

How does this incorporate the idea of asymptoticity?

Please kindly help.

It means that:
\lim_{x\to{x}_{0}}\frac{f(x) - \sum_{n=0}^N a_n(x-x_0)^n}{(x-x_{0})^{N}}=0