Calculus II · Unit 4B · lesson

Power Series as Functions

Concept

Learning objectives

interpret a power series as a function and evaluate it at specific inputs.

Power Series as Functions

Explanation

An infinite polynomial is still only meaningful where it converges

A power series resembles a polynomial with infinitely many terms. For each fixed input $x$ , however, it becomes an ordinary numerical series. Its value exists only if that numerical series converges. The same symbolic expression may therefore define a function on an interval and fail completely outside it.

The center $a$ organizes the powers $(x-a)^n$ . Near the center, these powers are small and convergence is favored. Farther away, they grow and can overwhelm the coefficients. Evaluating a power series begins by substituting the input, simplifying the resulting numerical series, and then using the convergence tools from Unit 4A.

Bridge

One infinite formula can define an entire function

A numerical series asks whether one fixed list of numbers has a finite total. A power series inserts a variable into the terms, so convergence can change when $x$ changes. The same expression may converge rapidly at one input, converge delicately at another, and diverge immediately elsewhere.

This is why a power series should be viewed as both an infinite series and a function. First choose an input $x$ ; then the power series becomes an ordinary numerical series. The set of inputs where that numerical series converges is the function's domain of representation.

**A power series changes character when x changes**
Three panels show geometric partial sums for x=1/2 approaching 2, for x=-1 oscillating, and for x=2 growing rapidly.
A power series changes character when x changes. Partial sums of the geometric series at an interior, endpoint, and exterior input.

Definition

Power series centered at a

A power series has the form

\sum_{n=0}^{\infty}c_n(x-a)^n.

For each fixed $x$ , this becomes a numerical series whose convergence must be determined.

Concept

Power-series form

A power series centered at $a$ is

\sum_{n=0}^{\infty}c_n(x-a)^n.

At $x=a$ , every positive-power term vanishes, so the value is $c_0$ .

Guided walkthrough

Evaluate at several inputs

For

F(x)=\sum_{n=0}^{\infty}\left(\frac{x}{2}\right)^n,

we have $F(0)=1$ , $F(1)=\sum(1/2)^n=2$ , and at $x=3$ the terms $(3/2)^n$ do not approach zero, so the series diverges.

Worked example

The same power series behaves three different ways

Consider

\sum_{n=0}^{\infty}x^n.

At $x=1/2$ , it is geometric with ratio $1/2$ and sums to $2$ . At $x=-1$ , the terms alternate between $1$ and $-1$ , so they do not approach zero and the series diverges. At $x=2$ , the terms grow and the series diverges even more plainly. The expression therefore defines $1/(1-x)$ only for $|x|<1$ .

Common mistake

An algebraic formula does not erase the convergence domain

The identity $\sum x^n=1/(1-x)$ is valid where the geometric series converges, not at every point where the rational function itself exists.

Interactive checku4b-power_series_as_functions-01

Evaluate $\sum_{n=0}^{\infty}(1/2)^n$ .

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Show hint

It is geometric with first term $1$ and ratio $1/2$ .

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Exercise

Evaluate $\sum(x/3)^n$ at $x=0,1,4$ .

Exercise

Identify the center of $\sum c_n(x+2)^n$ .

Exercise

Explain why a power series is not automatically defined for every real $x$ .

Exercise

Find the value at the center without summing.

After the explanation

Use the section idea

Reading lens

Find the radius from interior behavior, then test each boundary point as a separate numerical series.

Mental model

Distance from the center organizes the automatic interior and exterior behavior; endpoints remain independent decisions.

Decision

Use a ratio or root argument for the radius, convert it to an interval, and test both endpoints explicitly.

Common trap

Including or excluding both endpoints from the radius calculation alone.

Check yourself

Can you justify the radius and each endpoint with separate evidence?

Source & rights

Original instruction with traceable references.

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