Definition of asymptotic relation


by hanson
Tags: asymptotic, definition, relation
hanson
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#1
Jul19-06, 10:19 AM
P: 320
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,

[tex]
f(x) - \sum_{n=0}^N a_n(x-x_0)^n << (x-x_0)^N
[/tex]

How does this incorporate the idea of asymptoticity?

Please kindly help.
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arildno
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Jul19-06, 10:37 AM
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Quote Quote by 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,

[tex]
f(x) - \sum_{n=0}^N a_n(x-x_0)^n << (x-x_{0})^{N}
[/tex]

How does this incorporate the idea of asymptoticity?

Please kindly help.
It means that:
[tex]\lim_{x\to{x}_{0}}\frac{f(x) - \sum_{n=0}^N a_n(x-x_0)^n}{(x-x_{0})^{N}}=0[/tex]


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