Definition of asymptotic relation

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SUMMARY

The asymptotic relation between a function and a power series is defined by the expression f(x) - ∑_{n=0}^N a_n(x-x_0)^n << (x-x_0)^N. This indicates that as x approaches x_0, the limit of the difference between the function and its power series expansion, normalized by (x-x_0)^N, approaches zero. This definition incorporates the concept of asymptoticity by demonstrating that the error in the approximation becomes negligible compared to the growth of (x-x_0)^N as x approaches x_0.

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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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hanson said:
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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