Power series, formal power series and asymptotic series

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  • #1
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What's the difference between these three?
I only know Taylor series and its variants which I suppose is called power series (but I'm not sure). In that you just approximate around a single point using derivatives.

But what are formal powers series and asymptotic expansion?
I did see Wikipedia but all seem to be similar, I don't find anything very different among these three.
 

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  • #2
pasmith
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What's the difference between these three?
I only know Taylor series and its variants which I suppose is called power series (but I'm not sure). In that you just approximate around a single point using derivatives.

But what are formal powers series and asymptotic expansion?
I did see Wikipedia but all seem to be similar, I don't find anything very different among these three.

Formal power series are expressions of the form
[tex]
\sum_{n = 0}^\infty a_n x^n
[/tex]
where the [itex]a_n[/itex] are real (or complex) and [itex]x[/itex] is just a symbol and does not denote a real or complex number. The ring of formal power series under obvious operations of addition and multiplication using the rule [itex]x^nx^m = x^{n + m}[/itex] is isomorphic to the ring of real (or complex) sequences under appropriate definitions of addition and multiplication; if [itex]a : \mathbb{N} \to \mathbb{R}[/itex] and [itex]b : \mathbb{N} \to \mathbb{R}[/itex] are sequences then
[tex]
(a + b)_n = a_n + b_n, \\
(a \times b)_n = \sum_{k=0}^n a_k b_{n-k}.
[/tex]

"Power series" are what you get if you take one of the above and decide that [itex]x[/itex] is actually a real (or complex) number, rather than a symbol. Then you have problem of determining whether you can actually do the sum, and we define
[tex]
\sum_{n = 0}^{\infty} a_n x^n = \lim_{N \to \infty} \sum_{n = 0}^N a_n x^n
[/tex]
for all [itex]x \in \mathbb{R}[/itex] such that the limit exists. Taylor series are a special case of power series.

Asymptotic series are entirely different, and we say that [itex]\sum_{n=0}^N a_n x^n[/itex] is asymptotic to [itex]f(x)[/itex] (written [itex]\sum_{n=0}^N a_n x^n \sim f(x)[/itex]) as [itex]x \to 0[/itex] if and only if for all [itex]M \leq N[/itex],
[tex]
\lim_{x \to 0} \frac{\sum_{n=0}^M a_n x^n - f(x)}{x^M} = 0
[/tex]

The idea is that a truncated series [itex]\sum_{n=0}^N a_n x^n[/itex] can provide a useful approximation to [itex]f[/itex] even when the full series [itex]\sum_{n=0}^\infty a_n x^n[/itex] does not converge.
 
  • #3
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So when I write ex = 1 + 1/x + 1/x2 + 1/x3 + ...
which type of series is this?
It is also a formal power series as x is just a symbol.

One more question: If I try to find all these for a function, does it require different techniques?
 

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