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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?

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Sorry I don't know how to format, someone please help me about the summation. Thanks a lot. – LLS Feb 14 '11 at 10:52
to address your first question: A series from $- \infty$ to $\infty$ is called Laurent series. – Rudy the Reindeer Feb 14 '11 at 10:53
You enclose latex between dollar signs, to edit your post you click on 'edit' in the bottom left corner of your post. – Rudy the Reindeer Feb 14 '11 at 10:54
@Matt: Thank you. But is Laurent series also a kind of power series? (Like Taylor series is power series.) – LLS Feb 14 '11 at 10:55
I don't think the term "power series" is rigorously defined. For instance, the Wikipedia article on "power series" assumes non-negative integer powers, but the article on the Laurent series calls it a "power series" all the same. Any series of powers of something can be called a "power series". Theorems are usually proved for certain kinds of power series, e.g. Taylor series and Laurent series, or for series in general; if you come across a theorem that claims to apply to "power series", you should probably look at its proof to decide whether it assumes non-negative integer powers. – joriki Feb 14 '11 at 11:20
up vote 5 down vote accepted

"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.

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Why non-negative integer coefficients? I'd say 'generally' (such as the ones encountered in real and complex analysis) the coefficients are real or complex numbers? – wildildildlife Feb 14 '11 at 22:03
Er, exponents. Sorry. – Qiaochu Yuan Feb 14 '11 at 22:21
Thanks. Actually, this question has been bothering me since I learned the Laurent series because the book only gives theorems in the form of non-negative powers. And the author may naturally think there is no difference. The only thing I could find is that uniformly converged series can be differentiated termwisely. Now in the study of differential equation, the fraction power appears, and I have no idea which theorems can be used. Can you give me some guidance? – LLS Feb 15 '11 at 13:50
@LLS: that depends. There's a big difference between exponents with denominators dividing a fixed positive integer (since you can make a simple substitution to deal with this) and exponents with denominators which get arbitrarily large in lowest terms. I don't know anything about convergence properties of the latter. – Qiaochu Yuan Feb 15 '11 at 14:47
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