Calculus II · Unit 4B · lesson

The Geometric Series as a Function Library

Concept

Learning objectives

produce new power-series identities by substitution, differentiation, integration, and algebra.

The Geometric Series as a Function Library

Explanation

One formula generates a surprising amount of calculus

The identity $1/(1-x)=\sum x^n$ is the seed from which many useful power series grow. Replacing $x$ by another expression shifts or rescales the center. Differentiation creates coefficients involving $n$ . Integration creates denominators involving $n+1$ . Algebra combines these into rational-function expansions.

Every transformation carries a convergence condition. If $x$ is replaced by $u(x)$ , require $|u(x)|<1$ before endpoint analysis. The best workflow is to write the starting identity and interval, perform one transformation at a time, and update the condition explicitly.

Bridge

One identity generates a library of functions

The geometric identity

\frac1{1-x}=\sum_{n=0}^{\infty}x^n

is the seed from which many power series can be built. Replacing $x$ , multiplying by a factor, differentiating, or integrating transforms the known series into new ones.

The transformation must be applied consistently to the function, every term, and the convergence condition. A substitution such as $x\mapsto -x^2$ changes both powers and the domain $|x|<1$ . Keeping a visible operation pipeline prevents dropped signs and incorrect intervals.

**The geometric series generates a family of expansions**
A central geometric-series identity branches through substitution, scaling, differentiation, and integration to several new function series.
The geometric series generates a family of expansions. A transformation tree branching from the geometric series.

Concept

Geometric seed

For $|x|<1$ ,

\frac1{1-x}=\sum_{n=0}^{\infty}x^n.

Guided walkthrough

Expand a shifted rational function

Write

\frac1{3-x}=\frac13\frac1{1-x/3} =\frac13\sum_{n=0}^{\infty}\left(\frac{x}{3}\right)^n =\sum_{n=0}^{\infty}\frac{x^n}{3^{n+1}},

valid for $|x|<3$ .

Worked example

Transform the geometric series systematically

Find a series for

\frac{3}{2+x}.

Rewrite

\frac{3}{2+x}=\frac32\frac1{1+x/2} =\frac32\frac1{1-(-x/2)}.

Therefore

\frac{3}{2+x}=\frac32\sum_{n=0}^{\infty}\left(-\frac{x}{2}\right)^n, \qquad |x|<2.

The interval follows from $|-x/2|<1$ .

Common mistake

Rewrite into 1 over 1 minus u before expanding

Guessing signs from the denominator is unreliable. Explicitly identify $u$ in $1/(1-u)=\sum u^n$ , then transform the convergence condition $|u|<1$ .

Interactive checku4b-geometric_series_library-01

Find a power series for $1/(3-x)$ centered at zero.

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Show hint

Factor out $3$ and match $1/(1-r)$ .

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Exercise

Expand $1/(1+x^2)$ .

Exercise

Expand $1/(2+x)$ around zero.

Exercise

Differentiate to expand $1/(1-x)^2$ .

Exercise

State the convergence condition after substituting $x^2$ for $x$ .

After the explanation

Use the section idea

Reading lens

Treat algebra, differentiation, and integration as coefficient transformations with a preserved or rechecked interval.

Mental model

Inside a power series' convergence interval it behaves like a polynomial limit, so familiar operations become structured index shifts.

Decision

Write the source identity and validity interval before transforming coefficients or powers.

Common trap

Losing an index, constant of integration, or endpoint condition during a formal manipulation.

Check yourself

Can you reverse the operation and recover the source series?

Source & rights

Original instruction with traceable references.

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