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Suppose a continuous function $F(\lambda )$ on $[0,\infty )$ has a known power-law tail $F(\lambda )\sim a{\lambda }^{-p}$. Put $u=\lambda /(1+\lambda )$ and $h(u)=(1+\lambda {)}^{p}F(\lambda )$. When $h$ extends continuously to $u=1$, polynomial approximation on $[0,1]$ gives a global approximation $F(\lambda )\approx (1+\lambda {)}^{-p}{h}_N(u)$ using only elementary operations. Taylor series at zero and the tail asymptotic expansion can constrain endpoint coefficients. Agreement at the two endpoints alone does not guarantee accuracy in the intervening interval; uniform error control or intermediate calibration is needed.

For the degree-$N$ partial sum $P_N(z)=1+z+\cdots +z^N$ of the geometric series, the finite identity $(1-z)P_N(z)=1-z^{N+1}$ givesFor any fixed $0<r<1$, the triangle inequality then gives the uniform error boundEquality is attained at the positive real point $z=r$, so this is the exact worst-case error on that closed disk. It also proves uniform convergence on each such disk, not just pointwise convergence.

Given a positive tolerance $\epsilon$, choosing $N$ with $r^{N+1}/(1-r)\le \epsilon$ turns a convergence statement into a finite stopping rule. For $\epsilon =10^{-3}$, degree ten suffices throughout $|z|\le 1/2$. Nearer the boundary, $r=0.9$ requires degree $87$ and $r=0.99$ requires degree $1145$. The required degree depends on both the tolerance and the region, not just on the name of the function.

An elementary geometric-series construction suffices, without assuming a general polynomial-approximation theorem. Rotate the square by a multiple of $\pi /2$ so that $\Re w>1$. Put $c=-R$ with $R>0$. Every point of the square then satisfies $|z-c{|}^2\le (R+1{)}^2+1$, whereasfor sufficiently large $R$. Thus the whole square lies inside a disk centred at $c$ whose radius is less than $|w-c|$. The other three ways $w$ can lie outside the square are handled by the indicated rotations, after which rotating back preserves polynomial form.

Now expandLet $\rho ={\max }_{z\in K}|z-c|/|w-c|<1$. The tail after degree $m$ is uniformly bounded by ${\rho }^{m+1}/[|w-c|(1-\rho )]$, tending to zero. Hence the displayed partial sums are the required uniform polynomial approximations. This is the Taylor series about a carefully chosen distant centre, rather than a potentially inadequate series about $0$.

Use the unscaled Finite Fresnel integrals of this paper. Termwise integration of the absolutely convergent Taylor series for sine and cosine gives the full small-argument expansionsIn particular,These Taylor series converge at every finite argument, although cancellation makes them unattractive for large arguments.

For the large-argument asymptotic expansions, combine the two Finite Fresnel integrals into $C+iS$. The standard Fresnel integral givesFor the tail $J(\lambda )={\int }_{\lambda }^{\infty }{e}^{i{t}^2}dt$, integration by parts using $(e^{i{t}^2}{)}^{\prime }=2it{e}^{i{t}^2}$ givesFor example $a_1=1/4$, $a_2=-3i/8$, and $a_3=-15/16$. Repeated integration by parts bounds the remainder after any fixed number of terms by the order of the next term. Equivalently the Fresnel tail expansion isHere $(-1)!!=1$. The first three terms of the requested cosine Fresnel integral, counting its constant term, areFor comparison, retaining one further correction in both Finite Fresnel integrals givesUnlike the small-argument Taylor series, these large-argument asymptotic expansions must be truncated; adding indefinitely many terms at a fixed argument does not improve the approximation.

A calculator needs elementary arithmetic and its built-in sine and cosine, together with stored approximation coefficients. A convenient global construction uses compactified approximation from endpoint asymptotics. Define the exact real tail functions byThen $P(0)=Q(0)=A$, while $P\sim 1/(2\lambda )$ and $Q\sim 1/(4{\lambda }^3)$ at infinity. Consequentlyextend continuously to the compact interval $0\le u\le 1$, with $p(1)=1/2$ and $q(1)=1/4$. Choose stored polynomial approximations $p_N,q_N$ on that interval, constrained by the small-argument Taylor series and large-argument asymptotic expansions, and calibrated across the intervening interval. The resulting elementary formula isUniform errors ${\delta }_p,{\delta }_q$ in the stored polynomial approximations give $|C_{\mathrm{calc}}-C|\le {\delta }_p+{\delta }_q$ for every positive argument. Existence of arbitrarily accurate such approximations follows from the Weierstrass approximation theorem. A Padé approximant with a denominator having no positive zero is an alternative implementation. Matching the two end expansions alone is not an error guarantee: the middle interval must also be controlled. For very small arguments, switching to the short Taylor series avoids cancellation. Ordinary finite-precision limitations in evaluating ${\lambda }^2$ and reducing its trigonometric phase remain those of the calculator itself.

Starting with $p_0=0$ on $[-1,1]$, this polynomial recurrence gives $0\le p_r(x)\le |x|$ and $p_{r+1}(x)\ge p_r(x)$. Indeed the error satisfies $|x|-p_{r+1}=(|x|-p_r)[1-(|x|+p_r)/2]$. Its pointwise limit solves $p^2=x^2$, so it is $|x|$. Dini's theorem gives uniform convergence. Each continuous piecewise linear function is an affine function plus a finite linear combination of $(x-t{)}_{+}=(x-t+|x-t|)/2$. Scaling and translating the recurrence therefore gives polynomial approximation of every such function, and their density gives the Weierstrass approximation theorem.

The radius of convergence of ${\sum }_{n\ge 0}{a}_n(z-c{)}^n$ is the unique $R\in [0,\infty ]$ separating interior absolute convergence from exterior divergence. It converges absolutely when $|z-c|<R$ and diverges when $|z-c|>R$. At the centre it always has the finite value $a_0$, since all positive powers vanish. The boundary convergence of a power series is a separate question.

For $0<R<\infty$, the interior is a disk in the complex plane and an open interval $(c-R,c+R)$ on the real line. A power series converges uniformly on every smaller closed disk $|z-c|\le r<R$. This gives stable polynomial approximation over a whole compact region, rather than a separate limiting calculation at each point. See Jiří Lebl, Basic Analysis, §11.3.2: convergence of power series.

Four examples reveal what $R$ measures:

The Cauchy-Hadamard theorem computes $R$ directly from the coefficients. The attractive feature is that one growth statistic decides convergence throughout the interior and exterior, including sparse series with many zero coefficients. Finite changes to the coefficients do not alter the radius, though they can change the sum.