Power series, formal power series and asymptotic series

[Avichal]

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.

 

[pasmith]

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
$$\sum_{n = 0}^\infty a_n x^n$$
where the $a_n$ are real (or complex) and $x$ 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 $x^nx^m = x^{n + m}$ is isomorphic to the ring of real (or complex) sequences under appropriate definitions of addition and multiplication; if $a : \mathbb{N} \to \mathbb{R}$ and $b : \mathbb{N} \to \mathbb{R}$ are sequences then
$$(a + b)_n = a_n + b_n, \\ (a \times b)_n = \sum_{k=0}^n a_k b_{n-k}.$$

"Power series" are what you get if you take one of the above and decide that $x$ 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
$$\sum_{n = 0}^{\infty} a_n x^n = \lim_{N \to \infty} \sum_{n = 0}^N a_n x^n$$
for all $x \in \mathbb{R}$ such that the limit exists. Taylor series are a special case of power series.

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

The idea is that a truncated series $\sum_{n=0}^N a_n x^n$ can provide a useful approximation to $f$ even when the full series $\sum_{n=0}^\infty a_n x^n$ does not converge.

 

[Avichal]

So when I write e^x = 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?