Power Series: Definitions, Convergence, and Applications

Section 9.7 Power Series

Definition of Power Series

A power series is an infinite series of the form:

• About x = 0 (Maclaurin series): $\sum_{n=0}^{\infty} c_n x^n$

• About x = a (Taylor series): $\sum_{n=0}^{\infty} c_n (x - a)^n$

Here, a is the center of the series, and the cn are constant coefficients. Power series are used to represent functions as infinite polynomials, which can be useful for approximation and analysis.

Polynomial Approximations of Power Series

Partial sums of a power series (polynomial approximations) can be used to approximate the function represented by the series. As more terms are included, the approximation improves within the interval of convergence.

• y0 = 1 (constant term)

• y1 = 1 + x (linear approximation)

• y2 = 1 + x + x^2 (quadratic approximation)

• ... and so on, up to higher-degree polynomials.

Example: The function $y = \frac{1}{1-x}$ can be represented by the power series $1 + x + x^2 + x^3 + \cdots$ for $|x| < 1$.

Convergence of Power Series

A power series may converge (sum to a finite value) for some values of x and diverge for others. The set of x values for which the series converges is called the interval of convergence.

• Absolute Convergence: The series $\sum c_n (x-a)^n$ converges absolutely if $\sum |c_n (x-a)^n|$ converges.

• Radius of Convergence (R): There exists a number R such that the series converges absolutely for $|x-a| < R$ and diverges for $|x-a| > R$.

• At the endpoints $x = a \pm R$, convergence must be checked separately.

Theorem 18: Convergence Theorem for Power Series

If the power series $\sum c_n (x-a)^n$ converges at $x = c$, then it converges absolutely for all $x$ with $|x-a| < |c-a|$. If it diverges at $x = d$, then it diverges for all $x$ with $|x-a| > |d-a|$.

Corollary to Theorem 18: Radius and Interval of Convergence

The convergence of a power series is described by one of three cases:

• There is a positive number R such that the series diverges for $|x-a| > R$ but converges absolutely for $|x-a| < R$.

• The series may or may not converge at the endpoints $x = a \pm R$.

• The series converges absolutely for every $x$ (i.e., $R = \infty$).

• The series converges only at $x = a$ ($R = 0$).

How to Test a Power Series for Convergence

To determine where a power series converges:

1. Use the Ratio Test (or Root Test) to find the largest open interval where the series converges absolutely.

2. If the radius of convergence R is finite, test for convergence or divergence at each endpoint using the Comparison Test, Integral Test, or Alternating Series Test.

3. If $|x-a| > R$, the series diverges because the nth term does not approach zero.

Theorem 19: Series Multiplication for Power Series

If two power series $\sum a_n (x-a)^n$ and $\sum b_n (x-a)^n$ converge absolutely for $|x-a| < R$, then their product also converges absolutely to the product of the sums for $|x-a| < R$.

Theorem 20: Continuity and Power Series

If $\sum c_n (x-a)^n$ converges absolutely for $|x-a| < R$ and $f$ is a continuous function, then $f(\sum c_n (x-a)^n)$ converges absolutely on the set of points $x$ where $|x-a| < R$.

Theorem 21: Term-by-Term Differentiation Theorem

If $\sum c_n (x-a)^n$ has radius of convergence $R$, then it defines a function $f(x)$ on $|x-a| < R$ with derivatives of all orders inside the interval. The derivatives can be found by differentiating the series term by term:

• $f'(x) = \sum_{n=1}^{\infty} n c_n (x-a)^{n-1}$

• $f''(x) = \sum_{n=2}^{\infty} n(n-1) c_n (x-a)^{n-2}$

• ... and so on.

Each derived series converges at every point of the interval $|x-a| < R$.

Theorem 22: Term-by-Term Integration Theorem

If $\sum c_n (x-a)^n$ converges for $|x-a| < R$, then the series obtained by integrating term by term also converges for $|x-a| < R$:

• $\int \sum_{n=0}^{\infty} c_n (x-a)^n dx = C + \sum_{n=0}^{\infty} \frac{c_n}{n+1} (x-a)^{n+1}$

Summary Table: Possibilities for Interval of Convergence

Case	Interval of Convergence	Description
(a)	None	Diverges everywhere except at center
(b)	[x-a, x+a]	Converges on closed interval
(c)	(x-a, x+a]	Converges on half-open interval
(d)	[x-a, x+a)	Converges on half-open interval
(e)	All real numbers	Converges everywhere
(f)	x = a	Converges only at center