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We were informally introduced Taylor series in my physics class as a method to give an equation of the electric field at a point far away from a dipole (both dipole and point are aligned on an axis). Basically for the electric field: $$\vec E_{axis}=\frac q {4πε_o}[\frac {1} {(x-\frac s 2)^2}- \frac {1} {(x+\frac s 2)^2}]$$

Where 's' is the separation of the dipole and ##(x\pm\frac s 2)^2## or ##r\pm##are the distances between the dipole and the point at 'x' distance away (you can see that one charge will be further away to 'x' than the other).

Well we learned that if ##r\pm\gg s##, then the equation becomes just: $$\vec E_{axis}=\frac 1 {4πε_o}\frac {2qs} {r^3}$$ We used taylor series to explain this phenomena.

What are you actually approximating in a taylor series, and why is it even useful if you have the equation in the first place? What's so special about wanting to choose a random point on a graph and make it look like the graph itself?

Also I don't understand how you can apply this in physics, for example the dipole derivation, and why it works. To me without using a taylor series, it made sense that if the radius is much larger than the distance between the dipoles, then obviously the distances between the dipoles would be so insignificant for the electric field that you don't even have to include it.

Where 's' is the separation of the dipole and ##(x\pm\frac s 2)^2## or ##r\pm##are the distances between the dipole and the point at 'x' distance away (you can see that one charge will be further away to 'x' than the other).

Well we learned that if ##r\pm\gg s##, then the equation becomes just: $$\vec E_{axis}=\frac 1 {4πε_o}\frac {2qs} {r^3}$$ We used taylor series to explain this phenomena.

**I understood the mathematics of taylor series, and I get the normal definition**, my problem is I'm struggling to understand how a taylor series*actually*works and what the point of it is.What are you actually approximating in a taylor series, and why is it even useful if you have the equation in the first place? What's so special about wanting to choose a random point on a graph and make it look like the graph itself?

Also I don't understand how you can apply this in physics, for example the dipole derivation, and why it works. To me without using a taylor series, it made sense that if the radius is much larger than the distance between the dipoles, then obviously the distances between the dipoles would be so insignificant for the electric field that you don't even have to include it.

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