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,

$$f(x) - \sum_{n=0}^N a_n(x-x_0)^n << (x-x_0)^N$$

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,

$$f(x) - \sum_{n=0}^N a_n(x-x_0)^n << (x-x_{0})^{N}$$

How does this incorporate the idea of asymptoticity?

Please kindly help.

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

 

[tacman]

The asymptotic relation between a function and a power series is defined in this way because it takes into account the behavior of the function as x approaches a certain value, typically denoted as x_0. The term "asymptotic" refers to the behavior of the function as it approaches a particular value, and this definition captures that by considering the difference between the function and the power series as x approaches a certain value.

The notation used, "<<" (much less than), indicates that the difference between the two expressions becomes increasingly smaller as x approaches x_0. In other words, the function and the power series approach each other in a way that is not equal, but the difference between them becomes insignificant as x gets closer to x_0. This is the essence of asymptoticity - the two expressions may not be exactly equal, but they behave similarly as x approaches a certain value.

Furthermore, the use of the summation symbol and the upper limit of N in the power series allows for a more precise definition of the asymptotic relation. By considering the sum of increasingly higher powers of (x-x_0), we can capture the behavior of the function at different orders of magnitude as x approaches x_0. This allows for a more comprehensive understanding of the asymptotic relationship between the function and the power series.

I hope this helps to clarify the definition of the asymptotic relation and how it incorporates the concept of asymptoticity.