Taylor Polynomials, Why only Integer Powers? So It seems that the definition of polynomial is that is is raised to an integer power, but why is this necessary? My question mainly arises from a proof of the solution to the Hydrogen atom in quantum mechanics, where they rely on the fact that the terminating term in series expansion of a function is an integer power. I know that Taylor series work without the need for decimal and fractional powers, but why not use them? Would it not work? Why are polynomials defined as only have integer powers, and why are decimal powers subpar?
 A: Polynomials are actually restricted to only have natural exponents, not integer. The importance of polynomials is that they form a large class of functions that we can actually compute precisely (given the accuracy of the coefficients) on any input (depending on the accuracy of the input). Functions we can compute are important and we like them, and that is why they are studied. If you can express something as a polynomial then you probably should do it. Of course, we can, and and do, define other kinds of functions. There is nothing bad about them, it's just they may not be polynomials. 
Now, when we are given a function that is not a polynomial, we might wish to approximate it by the simplest thing we can think of - a polynomial. Taylor polynomials do just that. They relate a function and its derivative (and of course repeated derivatives of a function can only be taken a number of times, counted by a natural number (though fractional derivatives exist)) to a polynomial. That polynomial may or may not approximate the function, depending on the function, but it's a start. If it works, then that's great. If it does not, then you look for other solutions. 
