Power series, formal power series and asymptotic series?

[imdow123]

What's the difference between these three? I read the Wikipedia article and all I understand is:-

Formal power series are infinite and do not necessarily converge

Power series converge

Asymptotic series is converge after finite steps

 

Am I right?

 

Also if we want to implement these (I mean using a computer), there are different algorithms for each? Why is that? All feel like they do the same thing but I read many algorithms to compute formal power series, asmyptotic expansion etc. Why?

[studiot]

 

Am I right?

 

I'm sorry to tell you that you are not right.

 

It would be very helpful to link to the article in question. Are you thinking of real or complex variables?

 

Power series have the form

[math]P(z) = {a_0} + {a_1}{z_1} + ... = \sum {_0^\infty {a_n}{z^n}} [/math]

They may be convergent or divergent or convergent within some 'radius of convergence'.

 

Asymptotic series have the form

[math]A(z) = {a_0} + \frac{{{a_1}}}{z} + \frac{{{a_2}}}{z} + ... = \sum {_0^\infty {a_n}{z^{ - n}}} [/math]

They do not converge after a finite number of terms, they are also infinite series.

However as each successive term involves a higher power of the the variable the series get closer and closer to the desired function as n tends to infinity.

That is the difference between the asymptotic expansion series and the desired function tends to zero as n tends to infinity.

 

If you compare the two forms you can see that a power series converges for |z| < 1, all other things being equal, whilst an asymptotic series will converge when |z| > 1

Edited February 9, 2014 by studiot

[imdow123]

Okay, maybe I misunderstood the article.

Then what is "formal" power series?

 

WIll the techniques differ in finding power series, asymptotic series and formal power series? By techniques, I mean "proper" algorithms used in computer science as I'm interested in implementing these. Thank You!

[studiot]

 

It would be very helpful to link to the article in question.

 

 

edit I noticed an typo in my asymptotic series, sorry

 

[math]A(z) = {a_0} + \frac{{{a_1}}}{z} + \frac{{{a_2}}}{{{z^2}}}.... = \sum {_0^\infty } {a_n}{z^{ - n}}[/math]

 

Edited February 9, 2014 by studiot

[John]

Okay, maybe I misunderstood the article.

Then what is "formal" power series?

 

The formal power series is a generalization of the polynomial.

 

Formally, a polynomial in a single indeterminate [math]x[/math] can be expressed as the following sum:

 

[math]\sum_{i=0}^{\infty}a_{i}x^{i}[/math]

 

where each [math]a_{i}[/math] is a number (or, generally, an element of a ring), and with the restriction that only finitely many of the [math]a_{i}[/math]'s can be nonzero. While we're used to thinking of polynomials in the context of polynomial functions or equations, where we substitute in various values for [math]x[/math] to evaluate or solve a function or equation respectively, in this formal definition [math]x[/math] is simply [math]x[/math], i.e. it doesn't represent anything but itself. So while we might solve the polynomial equation

 

[math]x+6=3x[/math]

 

and find that "[math]x=3[/math]," the polynomials [math]x+6[/math] and [math]3x[/math] themselves are simply not equal.

 

A formal power series in a single indeterminate [math]x[/math] has a similar definition, except that it allows for infinitely many nonzero coefficients. Since [math]x[/math] doesn't really represent anything except itself, questions of convergence and divergence are meaningless.

Edited February 9, 2014 by John

[Someguy1]

Okay, maybe I misunderstood the article.

Then what is "formal" power series?

 

WIll the techniques differ in finding power series, asymptotic series and formal power series? By techniques, I mean "proper" algorithms used in computer science as I'm interested in implementing these. Thank You!

A formal power series is an expression that you don't substitute back into. In other words you have some power series, and you can add it to some other power series; or you could multiply it by some other power series. So the set of all power series in (say) one variable, is itself an algebraic object. You're just manipulating expressions. But you're not plugging in values of x and taking limits. Does that make sense? For example you can do the same trick with formal polynomials. x+2 plus 3x+5 = 4x+7 as formal polynomials that have no meaning or interpretation beyond their syntax.

[imdow123]

Makes sense now.

 

I have another question. There seems to be different algorithms to find power series, asymptotic series and formal power series.

What does formal power series of a function mean? Won't that be same as power series?

 

Why are there different algorithms to compute each type of series?

[Dave]

It's been a while since I looked at this, but I'm not sure the phrase "formal power series of a function" is well defined. You can calculate the power series of a (real/complex) function, which will generally be valid within some radius of convergence. That power series can also be viewed as a formal power series, in which we don't care about things like convergence, by choosing an indeterminate variable and the same coefficients as the power series. They won't be 'equal' because we (obviously!) don't know what type of variable the indeterminate is.