Combining Taylor Series Algebraically

Why Combine Power Series?

A known Taylor series can do more than approximate its original function. When two functions have known series, their sum, difference, and product can often be represented using those same series. The key is to combine coefficients correctly and to keep track of where the resulting equality is valid.

Addition and subtraction are direct: coefficients of matching powers are added or subtracted. Multiplication requires more care. A term of degree \(n\) in a product can come from several pairs of terms, so its coefficient is a sum of products.

Adding and Subtracting Series

Suppose two power series centered at \(c\) represent \(f\) and \(g\) in intervals around \(c\):

Where both series converge to their respective functions, the sum and difference are represented by

Each equality is guaranteed wherever both original series converge. In particular, if their radii of convergence are \(R_f\) and \(R_g\), both operations are guaranteed for \(|x-c|<\min(R_f,R_g)\). This is a common open interval centered at \(c\). At an endpoint, the original series need separate consideration; convergence at an endpoint should not be inferred just from the radii.

Multiplying Series: The Cauchy Product

For multiplication, a coefficient cannot usually be obtained by multiplying only the two coefficients with the same index. For example, a term in \(x^3\) can come from an \(x\) term times an \(x^2\) term, an \(x^2\) term times an \(x\) term, or a constant term times an \(x^3\) term.

When \(|x-c|\) is strictly less than both radii of convergence, each original power series converges absolutely. The Cauchy product then converges to the product of their sums. Consequently, within that common interval,

Why the Product Rule for Coefficients Works

The reason for the coefficient formula is that multiplication pairs every term of the first series with every term of the second. A pair \(a_k(x-c)^k\) and \(b_j(x-c)^j\) contributes \(a_kb_j(x-c)^{k+j}\). To contribute to degree \(n\), its indices must satisfy \(k+j=n\). The possible pairs are exactly \((0,n),(1,n-1),\ldots,(n,0)\), which gives the finite sum in the definition.

There is also a convergence issue: the infinite sums must be regrouped by total degree. Inside both radii of convergence, absolute convergence justifies that regrouping. For a fixed such \(x\), set \(u_k=a_k(x-c)^k\) and \(v_j=b_j(x-c)^j\). Then

Thus the double sum of products is absolutely convergent. Absolute convergence permits regrouping its terms by \(k+j=n\), without changing its value. Regrouping gives

Replacing \(u_k\) and \(v_j\) by their definitions proves that the Cauchy product represents the product of the original functions on the common open interval.

A Common Multiplication Mistake

A frequent error is to multiply only coefficients with matching powers, as if the coefficient of \(x^n\) in a product were \(a_nb_n\). For instance, this incorrect calculation would give

That misses the two ways to produce an \(x\) term: \(1\cdot x\) and \(x\cdot1\). The correct coefficient is \(1+1=2\), so the product begins \(1+2x+\cdots\). More generally, use \(d_n=\sum_{k=0}^{n}a_kb_{n-k}\), not just \(a_nb_n\).

Validity and Practical Use

For addition, subtraction, and multiplication, a safe general rule is to work inside the intersection of the original intervals of convergence. If the centers match and the radii are \(R_f\) and \(R_g\), this means \(|x-c|<\min(R_f,R_g)\). The resulting series may happen to converge farther, but that cannot be concluded merely from the original radii. Endpoints must be checked separately if they matter.

When finding only the first several terms of a product, compute each coefficient from the finite convolution sum. To find terms through degree \(N\), only coefficients \(a_k\) and \(b_j\) with \(k+j\leq N\) are needed. Do not confuse this finite calculation with saying the product is a polynomial: omitted higher-degree terms still belong to the full series.

These rules let a known collection of Taylor series describe many new functions without starting from derivatives again. The coefficient pattern also provides a useful check: every product coefficient must account for all pairs of terms that contribute to that power, and every series identity must be restricted to a region where the operations are justified.