Power Series and the Weierstrass Approximation Theorem

A power series centered at $a$ is a series of functions

n = 0 \sum \infty c_{n} (x - a)^{n},

whose partial sums are polynomials. The interchange theorems apply to it fully: inside its radius of convergence a power series converges uniformly on every closed subinterval, and so do all of its term-by-term derivatives. Term-by-term differentiation and integration are therefore valid, and the sum is an infinitely differentiable function.

Radius of convergence

A power series converges absolutely inside an interval and diverges outside it. The half-width of that interval is determined by how fast the coefficients grow.

Inside the interval $(a - ρ, a + ρ)$ the series converges pointwise. At the two endpoints it may do either, and the behavior there is a separate question. Between the center and either endpoint the convergence is uniform, which is what carries continuity and the calculus operations to the limit.

A power series converges absolutely on the open interval of radius

ρ

about

a

and diverges outside it; on each closed subinterval

[a - r, a + r]

with

r < ρ

the convergence is uniform.

The convergence is uniform on each closed subinterval but need not be uniform on the full open interval $(a - ρ, a + ρ)$ . The geometric series $\sum x^{n}$ has $ρ = 1$ and converges uniformly on every $[- r, r]$ with $r < 1$ , but not on $(- 1, 1)$ , where its sum $\frac{1}{1 - x}$ is unbounded near $1$ .

Term-by-term integration and differentiation

Uniform convergence on closed subintervals justifies integration term by term directly, since integration needs only uniform convergence of the functions.

Differentiation needs more: uniform convergence of the differentiated series. The key fact is that differentiating a power series does not change its radius, because $n^{1/ n} \to 1$ leaves the $lim sup$ untouched.

Applying the theorem to $f^{'}$ , then to $f^{''}$ , and so on, each derivative is again a power series with radius $ρ$ . A power series therefore defines an infinitely differentiable function on its interval of convergence.

Differentiating gives

\sum n c_{n} (x - a)^{n - 1}

and integrating gives

\sum \frac{c _{n}}{n + 1} (x - a)^{n + 1}

; both keep the radius

ρ

fixed, so the process iterates and every derivative is again a power series of radius

ρ

Analytic functions

A function that is locally the sum of a convergent power series has more structure than a merely smooth one.

Every analytic function is infinitely differentiable, but the converse fails: a function can be smooth and still not equal its Taylor series. The standard example is $g (x) = e^{- 1/ x^{2}}$ for $x \neq = 0$ and $g (0) = 0$ , which is infinitely differentiable with $g^{(n)} (0) = 0$ for all $n$ . Its Taylor series at $0$ is identically $0$ , yet $g (x) > 0$ for $x \neq = 0$ , so the series represents $g$ at no point other than $0$ . Being infinitely differentiable is weaker than being analytic.

The function

g (x) = e^{- 1/ x^{2}}

(with

g (0) = 0

) is smooth and positive off the origin, but every derivative vanishes at

0

, so its Taylor series there is identically

0

and matches

g

at no other point.

The Weierstrass approximation theorem

Power series show that some continuous functions are uniform limits of polynomials. The Weierstrass approximation theorem removes the qualifier: every continuous function on a closed bounded interval is a uniform limit of polynomials, whether or not it is analytic, and even where it is nowhere differentiable.¹

A constructive proof uses the Bernstein polynomials. Reduce to $[0, 1]$ by an affine change of variable, and for $f \in C [0, 1]$ define

B_{n} (x) = k = 0 \sum n f (\frac{k}{n}) (k n) x^{k} (1 - x)^{n - k} .

The weights $(k n) x^{k} (1 - x)^{n - k}$ are the probabilities of a binomial count, so $B_{n} (x)$ is a weighted average of the sampled values $f (k / n)$ , with the weight concentrated near $k / n \approx x$ . Two identities drive the proof:

The weights sum to one and have mean $x$ : $\sum_{k} (k n) x^{k} (1 - x)^{n - k} = 1$ and $\sum_{k} \frac{k}{n} (k n) x^{k} (1 - x)^{n - k} = x$ .
The spread is small. The variance identity $k = 0 \sum n (\frac{k}{n} - x)^{2} (k n) x^{k} (1 - x)^{n - k} = \frac{x ( 1 - x )}{n} \leq \frac{1}{4 n}$ forces the weight onto indices with $k / n$ near $x$ .

Given $ϵ > 0$ , uniform continuity of $f$ on the compact interval $[0, 1]$ gives $δ > 0$ with $∣ f (s) - f (t) ∣ < ϵ$ whenever $∣ s - t ∣ < δ$ . Split $B_{n} (x) - f (x) = \sum_{k} (f (\frac{k}{n}) - f (x)) w_{k} (x)$ into indices with $∣ \frac{k}{n} - x ∣ < δ$ , where each term is under $ϵ$ , and the rest, whose total weight is at most $\frac{1}{4 n δ ^{2}}$ by the variance bound and hence small for large $n$ . Both parts are controlled uniformly in $x$ , so $∥ f - B_{n} ∥_{[0, 1]} \to 0$ .

Bernstein polynomials

B_{n}

close in uniformly on a continuous target

f

[a, b]

; higher degree tightens the fit everywhere at once.

The construction is quantitative. The polynomials are named explicitly, and their degree $n$ controls the error through $\frac{1}{4 n δ ^{2}}$ , so a target accuracy dictates a concrete degree. The theorem also has a structural reading: $C [a, b]$ with the uniform norm is a complete metric space, and Weierstrass says the polynomials form a dense subset of it, so a function is continuous if and only if it is a uniform limit of polynomials.

Summary

Operation on $\sum c_{n} (x - a)^{n}$	Needs	Result
Continuity of the sum	uniform on $[a - r, a + r]$	continuous on $(a - ρ, a + ρ)$
Term-by-term $\int$	uniform convergence of terms	radius $\geq ρ$
Term-by-term $\frac{d}{d x}$	uniform convergence of derivatives	radius $= ρ$ , iterable
Analytic $\Rightarrow$ smooth	$c_{n} = f^{(n)} (a) / n!$	infinitely differentiable
Weierstrass density	uniform continuity on $[a, b]$	polynomials dense in $C [a, b]$

Uniform convergence, established by the M-test on closed subintervals, is the single hypothesis underneath the first three rows. The last two rows separate the cases: analytic functions are precisely those a series represents, and Weierstrass shows polynomials still approximate every continuous function as closely as desired.

Rosenlicht, Ch. 7 — Interchange of limit operations, treats uniform convergence and the approximation of continuous functions by polynomials on a closed bounded interval. The Bernstein-polynomial construction gives an explicit approximating sequence and an error controlled by the modulus of continuity of $f$ . ↩