Power Series from MTW 19.1: Explained?

In summary, a power series is an infinite series of the form ∑<sub>n=0</sub>∞ cn(x-a)<sup>n</sup>, used to represent functions in physics and make calculations easier. It differs from a regular series in that it has an infinite number of terms and can only represent certain types of functions. The convergence of a power series can be determined using the ratio test.
  • #1
zn5252
72
0
hello
Please see attachment which is a snapshot from MTW first page of chapter 19. Can someone please elaborate on how the equations 19.3b and c can be explained ?
I know that equation 19.3a is a familiar formula but not so much the other two. I'm just confused.
Thank you,
Long live & clear skies.
 

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  • #2
Look up "Multipole expansion"
 

FAQ: Power Series from MTW 19.1: Explained?

What is a power series?

A power series is an infinite series of the form ∑n=0∞ cn(x-a)n, where cn are constants, x is a variable, and a is a fixed point called the center.

How is a power series different from a regular series?

A regular series has a finite number of terms, while a power series has an infinite number of terms. Power series are also used to represent functions, while regular series are used for numerical calculations.

What is the purpose of using a power series in physics?

Power series are commonly used in physics to approximate functions and make calculations easier. They can also be used to represent physical phenomena such as motion, heat transfer, and electromagnetic fields.

How do you determine the convergence of a power series?

The convergence of a power series can be determined by using the ratio test, where the absolute value of the ratio of the (n+1)th term to the nth term is taken as n approaches infinity. If this ratio is less than 1, then the series is convergent. If it is greater than 1, then the series is divergent.

Can a power series represent any function?

No, not all functions can be represented by a power series. The function must be infinitely differentiable at the center point and have a radius of convergence, which is the distance from the center where the series converges. Some functions, such as those with discontinuities or singularities, cannot be represented by a power series.

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