The Taylor series of a smooth function (at a given point ) is a formal power series (in ) whose partial sum?s are the Taylor polynomials of (at ). As the Taylor polynomials are approximations of by polynomials (up to a given degree), so the Taylor series is an approximation of by an analytic function (or at least an asymptotic expansion that attempts to be this).
See also Taylor's theorem for error estimates in the convergence of Taylor series.
The Taylor series of at is the formal power series
If is analytic at , then the only power series witnessing this is the Taylor series of at (so in particular, the Taylor series exists; analytic functions are smooth).
In contrast, a smooth function need not be analytic; the classic counterexample is a bump function. In fact, the Taylor series of at might not converge to anywhere except at , either because the Taylor series has vanishing radius of convergence or because it converges to something else (an analytic function with the same jet as but a different germ). However, we can say this:
obtained by forming Taylor series in variables is surjective.
In particular, every power series in one real variable is the Taylor series of some smooth function on the real line (even if it has vanishing radius of convergence and so is not the Taylor series of any analytic function).
The proof is reproduced for instance in MSIA, I, 1.3
See more at Borel's theorem.
Examples of sequences of local structures
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||germ of a space||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||finite field||p-adic integers||localization at (p)||integers|
|Lie theory||Lie algebra||formal group||local Lie group||Lie group|
|symplectic geometry||Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|