I saw definitions and theorem about power series are in the form of $\sum_{k=0}^n a_k (x-x_0)^k$. And it definitely doesn't include negative or noninteger powers. Nevertheless, I saw the theorems like
- If the two series converge to the same value on some interval, then the corresponding coefficients are the same.
- The series can be differentiated or integrated termwisely.
be used without justification. I couldn't find relevant theorems.
My question is: are the series containing negative and noninteger power terms still called power series, and thus the theorems could apply? If not, do the theorems I mentioned as well as other common theorems like termwise multiplication (Cauchy product) hold?