Are negative or noninteger powers still power series?

Question

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

1. If the two series converge to the same value on some interval, then the corresponding coefficients are the same.
2. 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?

Answer 1

"Power series" does not have a fixed meaning. Generally, but not always, it means a series with non-negative integer exponents, and most theorems you have seen proved about power series are about this case. Many of these theorems extend to Laurent series with a fixed negative lower bound away from $x_0$, and many of them can be proven just by multiplying by the appropriate power of $x - x_0$ and using the corresponding theorem for ordinary power series. If you want more assurance than this, Laurent series are usually treated in books on complex analysis.

Answer 2

• A "formal power series" has only non-negative exponents
• A "Laurent series" allows negative exponents too
• A "Puiseux series" allows fractional exponents too (with restriction)