Why a Power Series Has a Unique Local Representation
An analytic function is represented near a point by a convergent power series. The Taylor Representation Theorem identifies the coefficients of such a representation from the derivatives at its center, and the Uniqueness of Power-Series Coefficients corollary established that two series representing the same function on an open interval around their common center have the same coefficients. Here we prove a more local criterion: values along any sequence of distinct points approaching the center already determine the coefficients.
The key is to examine the first coefficient at which a series does not vanish. Near the center, the corresponding power dominates all the higher powers. Consequently, a nonzero power series cannot have zeros accumulating at its center. This observation gives both a description of its zeros near the center and a uniqueness theorem for power-series representations.
The First Nonzero Coefficient
Consider a power series centered at \(a\), $$ F(x)=\sum_{n=0}^{\infty}c_n(x-a)^n, $$ with positive radius of convergence. If at least one coefficient is nonzero, there is a least index \(m\) for which \(c_m\ne0\). Factoring out \((x-a)^m\) leaves a power series whose value at \(a\) is \(c_m\). By continuity, that remaining factor stays nonzero sufficiently close to \(a\).
Proof. Choose \(r\) with \(0<r<R\). The power series converges absolutely at distance \(r\) from its center, so $$ M=\sum_{k=1}^{\infty}|c_{m+k}|r^k $$ is finite. For \(|h|<R\), define $$ H(a+h)=c_m+\sum_{k=1}^{\infty}c_{m+k}h^k. $$ This series converges for \(|h|<R\), and factoring out \(h^m\) gives \(F(a+h)=h^mH(a+h)\). To see that \(H(a+h)\) stays nonzero near \(h=0\), take \(|h|\le r\). Since \((|h|/r)^k\le |h|/r\) for \(k\ge1\), $$ \left|\sum_{k=1}^{\infty}c_{m+k}h^k\right| \le \sum_{k=1}^{\infty}|c_{m+k}||h|^k \le \frac{|h|}{r}M. $$ Choose $$ \delta=\min\left\{r,\frac{|c_m|r}{2(M+1)}\right\}>0. $$ For \(|h|<\delta\), the last bound is less than \(|c_m|/2\). Therefore $$ |H(a+h)|\ge |c_m|-\left|\sum_{k=1}^{\infty}c_{m+k}h^k\right|>\frac{|c_m|}{2}>0. $$ Thus \(F(a+h)=h^mH(a+h)\) has no zeros for \(0<|h|<\delta\). When \(m=0\), \(h^m=1\), so \(F\) has no zeros anywhere in that neighborhood. \(\square\)
This proof makes precise the idea that the first nonzero term controls the behavior near the center. The other terms do not have to be absent: their combined contribution to \(H\) simply remains too small to cancel \(c_m\) sufficiently close to the center.
Values Accumulating at the Center Determine the Coefficients
We can apply the isolation result to the difference of two series. Their difference is again a power series on the common interval of convergence. If the two sums agree at infinitely many distinct points approaching their common center, that difference has zeros accumulating at the center. The isolation theorem rules this out unless every coefficient of the difference is zero.
Proof. Let \(R\) be the smaller of the two radii of convergence. For \(|x-a|<R\), the difference is the power series $$ F(x)-G(x)=\sum_{n=0}^{\infty}(u_n-v_n)(x-a)^n. $$ It converges there because both original series converge absolutely at every such point. By hypothesis, this difference is zero at each \(x_j\). If any coefficient \(u_n-v_n\) were nonzero, there would be a least index \(m\) with \(u_m-v_m\ne0\). The Isolation of Zeros Near the Center Theorem would then give a \(\delta>0\) such that the difference has no zero satisfying \(0<|x-a|<\delta\). But \(x_j\to a\) and \(x_j\ne a\), so some \(x_j\) must satisfy \(0<|x_j-a|<\delta\), a contradiction. Hence \(u_n-v_n=0\) for every \(n\). The two series therefore have identical terms and the same sum wherever both converge. \(\square\)
The points of agreement need not fill an interval, and the functions being compared need not be introduced by formulas that are easy to differentiate. The sequence condition alone suffices, provided the points are distinct, approach the center, and lie in the series’ interval of convergence.
Worked Examples
Worked Example: A Sequence of Values Determines a Geometric Expansion
Suppose a power series \(P(x)=\sum_{n=0}^{\infty}d_nx^n\), with positive radius of convergence, satisfies $$ P(1/k)=\frac{1}{1+1/k} $$ for every integer \(k\ge2\). We determine all its coefficients. For \(|x|<1\), the finite geometric identity is $$ (1+x)\sum_{n=0}^{N}(-x)^n=1-(-x)^{N+1}. $$ As \(N\to\infty\), \((-x)^{N+1}\to0\), so $$ \frac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^nx^n. $$ Both \(P\) and this geometric series converge on some interval around \(0\). For all sufficiently large \(k\), \(1/k\) lies in that common interval, and their sums agree there by the stated condition. These are distinct points tending to \(0\). The Uniqueness from Values Accumulating at the Center Theorem therefore gives $$ d_n=(-1)^n\qquad(n\ge0). $$ In particular, the first coefficients are \(d_0=1\), \(d_1=-1\), and \(d_2=1\). The values along the sequence \(1/k\) leave no alternative power-series coefficients.
Worked Example: A Piecewise Function Cannot Have a Power Series at the Center
Define $$ f(x)= \begin{cases} x^2,&x\ge0,\\ 2x^2,&x<0. \end{cases} $$ Suppose, for contradiction, that a power series \(P(x)=\sum_{n=0}^{\infty}c_nx^n\) represents \(f\) on some open interval around \(0\). For every sufficiently large integer \(k\), \(1/k\) lies in that interval and is positive, so $$ P(1/k)=f(1/k)=\frac{1}{k^2}. $$ The polynomial \(Q(x)=x^2\) is a power series centered at \(0\), and \(Q(1/k)=1/k^2\) at those same points. Uniqueness from Values Accumulating at the Center implies that the coefficients of \(P\) and \(Q\) agree. Thus \(P(x)=x^2\) throughout their common interval of convergence.
But for any negative \(x\) in that interval, the assumed representation gives both \(P(x)=f(x)=2x^2\) and \(P(x)=x^2\). These equalities would imply \(2x^2=x^2\), hence \(x^2=0\), which is impossible for negative \(x\). The contradiction shows that \(f\) has no power-series representation on a neighborhood of \(0\). The one-sided values approaching the center already force the only possible series, and that series fails on the other side.
Worked Example: The Leading Power Describes Nearby Zeros
For \(|x|<1\), the geometric series gives $$ F(x)=\frac{x^4}{1+x^2} =x^4\sum_{n=0}^{\infty}(-1)^nx^{2n} =x^4-x^6+x^8-x^{10}+\cdots. $$ The first nonzero coefficient is the coefficient of \(x^4\), which is \(1\). Factoring out the leading power gives $$ F(x)=x^4H(x),\qquad H(x)=1-x^2+x^4-x^6+\cdots=\frac{1}{1+x^2}. $$ For real \(x\), \(1+x^2\ge1\), so \(H(x)>0\). Consequently \(F(0)=0\), while \(F(x)>0\) for every nonzero \(x\) with \(|x|<1\). The zero at the center has leading order four, and there are no other zeros in this interval. This explicit factorization illustrates the general theorem: once the leading power is removed, the remaining factor stays nonzero near the center.
Scope and a Common Pitfall
The location of the accumulating points matters. The uniqueness theorem requires them to approach the center, where the two series can be compared by the first nonzero coefficient of their difference. The same argument also applies if the sequence approaches another point strictly inside the common interval of convergence, after expressing the sums locally as power series centered at that point. It does not apply to a sequence approaching only an endpoint of convergence: the estimates in the isolation proof require a positive radius around the accumulation point on which the relevant series converges.
A second distinction is between a series having many zeros and its coefficients all being zero. The Isolation of Zeros Near the Center Theorem says that a nonzero series has no noncentral zeros sufficiently close to its center; it does not say that the series has no zeros elsewhere. For example, a series may vanish at its center because its first several coefficients are zero. In that case, the least nonzero coefficient identifies the power that factors out, while the remaining factor determines whether any additional nearby zeros are possible.
The uniqueness result is useful whenever function values are known at a sequence of points but a full interval of equality is unavailable. It also explains why formal coefficient comparisons are decisive: if two convergent power series really give the same local behavior along points approaching their center, even their earliest differing coefficient would create a contradiction. In particular, a power series cannot acquire a different local representation by changing its coefficients while keeping all those accumulating values fixed.
Check Your Understanding
Use the theorems and examples in this tutorial to answer the following questions.
- Why does a nonzero power series have a least index with a nonzero coefficient?
- In the isolation proof, why does the factor \(H(a+h)\) remain nonzero for sufficiently small \(|h|\)?
- What conditions on a sequence of points allow the Uniqueness from Values Accumulating at the Center Theorem to be applied?
- Why does agreement at the positive points \(1/k\) rule out a power-series representation of the piecewise function in the second worked example?
- Why is a sequence approaching only an endpoint of convergence not covered by the isolation proof?