Can f(x) Be Expressed as an Exponent of x When Reversing a Series Expansion?

[dst]

Can the sum of all antiderivatives equating each derivative of a term x, to units in x, be expressed as x^y or some other exponent, provided those sums are known? I have no idea if that made sense in mathematese, so a simpler way would be to ask:

Taking the equation for distance/displacement/elephants with respect to time:

s = ut + 1/2at^2 + 1/3bt^3 + ... 1/26zt^26

Or: http://img264.imageshack.us/img264/4659/28096134sz4.jpg

(no idea how to use tex, no images either? is that notation even correct?)

Can f(x) be expressed as an exponent of x, analytically?

 

[kanato]

What you're asking about is called a Taylor series:
http://mathworld.wolfram.com/TaylorSeries.html

 

[dst]

I guess I'm horrid at explaining.

Yes, it's a Taylor series, but can you reverse it and get f(x) from it?

Let's say you're calculating distance from a point, i.e. s = ut + 1/2(at^2) + 1/3(bt^3), if you know a and b, could you simplify it?

 

[HallsofIvy]

Since some power series can give horrendously complex functions, no, there is no (elementary) way to go from a known power series to the function it describes.

 

[dst]

Since some power series can give horrendously complex functions, no, there is no (elementary) way to go from a known power series to the function it describes.

That's the kind of answer I'm looking for (but didn't want to be the case).

Are there any examples of someone reversing an expansion to get a formula?

 

[rs1n]

That's the kind of answer I'm looking for (but didn't want to be the case).

Are there any examples of someone reversing an expansion to get a formula?

There are some examples, but they are often "textbook" problems designed to have solutions. As someone already explained, there is no general method. However, if you look up generating functions and power series you will find plenty of examples.