What's the Reasoning behind the Maclaurin Series? How did Maclaurin discover it?

[Abraham]

I've taken maths through calc 3.

I understand the Maclaurin series represents a function f(x) as a power series: $$\sum(c_{n}x^{n})$$

But how the heck did Maclaurin figure out that the series $$\sum(c_{n}x^{n})$$ could represent f(x)? I mean, that's clearly not obvious from inspection. I want to know how someone made this discovery.

 

[Jerbearrrrrr]

I don't think it came over night. And I'm sure a load of approximations (like sinx=x) already existed. At some point, someone clever is going to come along and try and get a better approximation, or study when/why approximations are possible.

 

[some_dude]

I think he noticed that, for a smooth function, for small values of $$\delta$$, $$f(\delta)$$ is approximately equal to $$f(0) + c\ \delta$$ for some constant $$c$$. Then if you consider the second derivative, and so on, you would expect to get better and better approximations. And then if you consider infinitely many linear approximations of the function you've uniquely described the original function and so you recover it.

 

[HallsofIvy]

Here's one way of looking at it- instead of asking for a polynomial of degree n that will be exactly equal to f(x) at n+1 distinct points (as the Lagrange interpolating polynomial is), as for a polynomial of degree n that will give, exactly, f(0), f'(0), f''(0), ..., $f^(n)(0)$. That is, the MacLaurin Polynomial, of degree n, gives the value of f(x) and its first n derivatives, exactly, at x= 0.