Operations on Power Series

Power series can be added, multiplied, differentiated, and integrated term by term within their interval of convergence, making them a powerful computational tool.

Term-by-Term Operations

Definition

Given two power series $f(x) = \sum a_n x^n$ and $g(x) = \sum b_n x^n$ with radii $R_f$ and $R_g$ respectively:

Sum: $(f + g)(x) = \sum (a_n + b_n) x^n$ with radius $\geq \min(R_f, R_g)$
Cauchy product: $(f \cdot g)(x) = \sum_{n=0}^\infty c_n x^n$ where $c_n = \sum_{k=0}^n a_k b_{n-k}$ with radius $\geq \min(R_f, R_g)$
Composition: $f(g(x))$ is a power series when $|g(x)| < R_f$

Theorem7.3Term-by-Term Differentiation and Integration

If $f(x) = \sum_{n=0}^\infty c_n x^n$ has radius of convergence $R > 0$ , then:

$f$ is differentiable on $(-R, R)$ and $f'(x) = \sum_{n=1}^\infty n c_n x^{n-1}$
$f$ is integrable and $\int f(x)\,dx = C + \sum_{n=0}^\infty \frac{c_n}{n+1} x^{n+1}$
Both derived series have the same radius of convergence $R$ .

Applications

ExampleDeriving new series from known ones

Starting from $\frac{1}{1-x} = \sum x^n$ :

Differentiate: $\frac{1}{(1-x)^2} = \sum_{n=1}^\infty n x^{n-1} = \sum_{n=0}^\infty (n+1)x^n$
Integrate: $-\ln(1-x) = \sum_{n=0}^\infty \frac{x^{n+1}}{n+1} = \sum_{n=1}^\infty \frac{x^n}{n}$
Substitute $x \to -x^2$ : $\frac{1}{1+x^2} = \sum_{n=0}^\infty (-1)^n x^{2n}$
Integrate the last: $\arctan x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1}$

Setting $x = 1$ in the arctangent series gives Leibniz's formula: $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$

Uniqueness

RemarkIdentity theorem for power series

If two power series $\sum a_n (x-a)^n$ and $\sum b_n (x-a)^n$ agree on an open interval containing $a$ , then $a_n = b_n$ for all $n$ . This identity theorem means the power series representation of a function (when it exists) is unique. It follows that the coefficients must be the Taylor coefficients: $c_n = f^{(n)}(a)/n!$ .

The algebra of power series, combined with term-by-term calculus, provides an efficient framework for computing limits, integrals, and solutions to differential equations that would be difficult to handle by other methods.