Why Taylor Coefficients are Forced

Why Coefficients Cannot Be Chosen Freely

A power series may look like a list of freely selected numbers multiplying successive powers of \(x-a\). But if the series represents a function near its center \(a\), those numbers are constrained: the function’s derivatives at \(a\) determine them exactly.

This is more than a convenient way to calculate coefficients. It is a uniqueness result. Two power series with positive radius of convergence cannot represent the same function on a neighborhood of their center while having different coefficients. Similarly, a polynomial of degree at most \(m\) is uniquely determined by its derivatives at \(a\) from order zero through order \(m\).

Conditions Matter

A power series centered at \(a\) has the form

Its radius of convergence \(R\) describes the open interval \(|x-a|<R\) on which it converges. In this tutorial, the coefficient formula concerns a series with positive radius \(R>0\) that represents \(f\) throughout a neighborhood of \(a\). In particular, there must be an open interval around \(a\) where the series converges and its sum equals \(f(x)\).

Pointwise agreement only at the center is not enough. Nor is agreement merely at all points where a series happens to converge if those points do not include a neighborhood of the center. The positive-radius condition lets us differentiate the series as a function near \(a\).

Why Differentiating Reveals One Coefficient

A power series can be differentiated term by term at every point strictly inside its radius of convergence. More precisely, if its radius is \(R>0\), the differentiated series converges on \(|x-a|<R\) and equals the derivative of the original series there. This result can be applied repeatedly; each resulting derivative series also has radius \(R\).

The reason term-by-term differentiation is valid is that power series and their differentiated series converge uniformly on every closed interval \(|x-a|\leq r\) with \(r<R\). The standard theorem on differentiating a uniformly convergent series of functions then applies when the derivative series converges uniformly there. For power series, absolute convergence on such smaller intervals provides that uniform convergence. Thus the differentiation is justified locally around the center, not merely performed formally.

Written as a single equation, the \(k\)-th derivative series is

When \(x=a\), the \(n=k\) term has exponent zero and contributes \(c_k k!\). All terms with \(n>k\) vanish. This proves the formula, including the case \(k=0\), where \(f(a)=c_0\).

Worked Example: Recovering Coefficients from a Power Series

Why Matching Derivatives Forces Polynomial Coefficients

The same reasoning works for polynomials, without any convergence issue. Write a polynomial \(P\) of degree at most \(m\) using powers centered at \(a\):

Taking \(k\) derivatives and evaluating at \(a\) leaves just the \(j=k\) term, so \(P^{(k)}(a)=k!b_k\). Therefore each coefficient is fixed by the matching derivative:

Consequently, if two polynomials of degree at most \(m\) have equal derivatives at \(a\) for every order \(0\) through \(m\), their coefficients in powers of \(x-a\) are equal, so the polynomials are identical. The degree bound matters: matching derivatives only through order \(m\) does not determine terms of higher degree.

Worked Example: Derivative Data Determine a Cubic

What a Finite List of Derivatives Does Not Determine

Matching derivatives through a given order determines coefficients only through that order. A term involving a higher power of \(x-a\) disappears when evaluated at \(a\) after fewer differentiations.

A Necessary Caution: Agreement at One Point Is Not Enough

The positive-radius condition in the coefficient theorem cannot be discarded. If a power series converges only at its center, there is no neighborhood on which it can be differentiated as a representation of the function. Agreement at the single convergence point places no restriction on the function’s derivatives there.

For a concrete counterexample, set \(a=0\) and \(c_n=n!\). The series is \(\sum_{n=0}^{\infty}n!x^n\). At \(x=0\), it has value \(1\). If \(x\neq 0\), the absolute values of successive terms have ratio

which eventually exceeds \(1\) and grows without bound. In particular, the terms do not approach zero, so the series diverges at every \(x\neq 0\). Its convergence set is only \(\{0\}\).

Now take \(f(x)=1+2x\). At the only point where the series converges, \(x=0\), it agrees with \(f\): both values are \(1\). But \(f'(0)=2\), whereas the series coefficient \(c_1\) is \(1\). There is no contradiction: the series does not represent \(f\) on a neighborhood, and its radius of convergence is zero.

A separate common error is to forget the factorial. For example, if the coefficient of \((x-a)^2\) is \(c_2\), then \(f''(a)=2!c_2\), not \(c_2\). The correct recovery is \(c_2=f''(a)/2!\).

Why This Result Matters

The coefficient formula explains why derivative information is not merely one possible way to build a power series. For a function represented by a power series around \(a\), each coefficient is compulsory. If two such series represent the same function on a neighborhood of \(a\), differentiating that shared function at \(a\) gives the same derivatives, and hence the same coefficients at every order.

For polynomials, the finite version is equally useful: values of the derivatives through order \(m\) specify exactly one polynomial of degree at most \(m\). But a finite list of derivatives cannot rule out additional higher-degree terms. Keeping these two claims distinct—and checking the positive-radius condition for an infinite series—prevents common misuses of the coefficient formula.