What is the connection between power series and Taylor series?

[rwx1606]

What exactly the connection between a power series and taylor series? I know all taylor series are power series with the coefficients = f^n(a)/n!, but what can we say about power series? My graduate student instructor said that all convergent power series are taylor series and I don't quite understand that. To me, taylor series just seems like a special case of power series where the coefficients has a special formula. Anyone care to explain? Thanks in advance.

 

[Dick]

I think you've already said it. All taylor series are power series. If the power series converges on a finite interval to some function f(x), then you can regard the power series as a taylor series for f(x). It's nothing really deep.

 

[toni]

so can we say taylor series represents function in a more precise way than power series does?

 

[Dick]

so can we say taylor series represents function in a more precise way than power series does?

If a power series 'represents' a function (by being equal to it on a open interval surrounding a point), then it IS the taylor series.

 

[Hurkyl]

What exactly the connection between a power series and taylor series? I know all taylor series are power series with the coefficients = f^n(a)/n!, but what can we say about power series? My graduate student instructor said that all convergent power series are taylor series and I don't quite understand that. To me, taylor series just seems like a special case of power series where the coefficients has a special formula. Anyone care to explain? Thanks in advance.

Exercise: Let
$$\sum_{n = 0}^{+\infty} a_n x^n$$
be a power series that converges for $x \in (-R, R)$. Find a function $f(x)$ whose Taylor series is that power series.

 

[HallsofIvy]

What exactly the connection between a power series and taylor series? I know all taylor series are power series with the coefficients = f^n(a)/n!, but what can we say about power series? My graduate student instructor said that all convergent power series are taylor series and I don't quite understand that. To me, taylor series just seems like a special case of power series where the coefficients has a special formula. Anyone care to explain? Thanks in advance.

Not just a "special case". If a power series converges to a function, the it is a Taylor's series for that function.

Problem: Find the Taylor's series for 1/(1- x) around x= 0 and find its radius of convergence. Yes, you could calculate the derivatives and evaluate at x= 0. That's not very difficult.

But it is much easier to recall that, if |r|< 1, then the geometric series
$$\sum_{n=0}^\infty a r^n= \frac{a}{1- r}$$
and just take a= 1, x= r: The Taylor's series for 1/(1- x) about x= 0 is
$$\sum x^n$$.
Because that is a power series that converges to 1/(1-x), it is the Taylor's series. Also, the fact that the geometric series converges for |r|< 1 and diverges for |r|> 1 tells us that the radius of convergence is 1.

 

[toni]

Exercise: Let
$$\sum_{n = 0}^{+\infty} a_n x^n$$
be a power series that converges for $x \in (-R, R)$. Find a function $f(x)$ whose Taylor series is that power series.

i donno...seems i still not getting the point at all ==;

it's like if I am given a question asking me to find a taylor series of a function, i will definitely lost. and the "center of convergence" how am i going to find it? using convergence test to find the interval of x and pick the middle point?