Editing Taylor's theorem (section)

=== Taylor expansions of real analytic functions ===

Let ''I'' ⊂ '''R''' be an [[open interval]]. By definition, a function ''f'' : ''I'' → '''R''' is [[analytic function|real analytic]] if it is locally defined by a convergent [[power series]]. This means that for every ''a''&nbsp;∈&nbsp;''I'' there exists some ''r''&nbsp;>&nbsp;0 and a sequence of coefficients ''c<sub>k</sub>''&nbsp;∈&nbsp;'''R''' such that {{nowrap|(''a'' − ''r'', ''a'' + ''r'') ⊂ ''I''}} and

<math display="block"> f(x) = \sum_{k=0}^\infty c_k(x-a)^k = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots, \qquad |x-a|<r. </math>

In general, the [[power series#Radius of convergence|radius of convergence]] of a power series can be computed from the [[Cauchy–Hadamard theorem|Cauchy–Hadamard formula]]

<math display="block"> \frac{1}{R} = \limsup_{k\to\infty}|c_k|^\frac{1}{k}. </math>

This result is based on comparison with a [[geometric series]], and the same method shows that if the power series based on ''a'' converges for some ''b'' ∈ '''R''', it must converge [[uniform convergence|uniformly]] on the [[closed interval]] <math display="inline">[a-r_b,a+r_b]</math>, where <math display="inline">r_b=\left\vert b-a \right\vert</math>. Here only the convergence of the power series is considered, and it might well be that {{nowrap|(''a'' − ''R'',''a'' + ''R'')}} extends beyond the domain ''I'' of ''f''.

The Taylor polynomials of the real analytic function ''f'' at ''a'' are simply the finite truncations

<math display="block"> P_k(x) = \sum_{j=0}^k c_j(x-a)^j, \qquad c_j = \frac{f^{(j)}(a)}{j!}</math>

of its locally defining power series, and the corresponding remainder terms are locally given by the analytic functions

<math display="block"> R_k(x) = \sum_{j=k+1}^\infty c_j(x-a)^j = (x-a)^k h_k(x), \qquad |x-a|<r. </math>

Here the functions

<math display="block">\begin{align}
& h_k:(a-r,a+r)\to \R \\[1ex]
& h_k(x) = (x-a)\sum_{j=0}^\infty c_{k+1+j} \left(x - a\right)^j
\end{align}</math>

are also analytic, since their defining power series have the same radius of convergence as the original series. Assuming that {{nowrap|[''a'' − ''r'', ''a'' + ''r'']}} ⊂ ''I'' and ''r''&nbsp;<&nbsp;''R'', all these series converge uniformly on {{nowrap|(''a'' − ''r'', ''a'' + ''r'')}}. Naturally, in the case of analytic functions one can estimate the remainder term <math display="inline">R_k(x)</math> by the tail of the sequence of the derivatives ''f′''(''a'') at the center of the expansion, but using [[complex analysis]] also another possibility arises, which is described [[Taylor's theorem#Relationship to analyticity##Taylor's theorem in complex analysis|below]].