Calculus II · Unit 4B · lesson

Building New Series from Known Ones

Concept

Learning objectives

use substitution, scaling, multiplication, differentiation, and integration to construct new power-series representations.

Building New Series from Known Ones

Explanation

Transformation is usually faster than repeated derivatives

Once a standard series is known, many new series can be generated by replacing the variable, multiplying by a power, rescaling, or combining identities. This approach exposes structure and reduces computational error. It also mirrors how power series are used in differential equations and special functions.

Every transformation must update the convergence condition. Substituting $x^2$ for $x$ changes $|x|<1$ to $|x^2|<1$ , which is still $|x|<1$ . Substituting $3x$ changes it to $|x|<1/3$ . Multiplication by a polynomial does not change the radius, but endpoints should still be checked when the representation is used there.

Bridge

Plan the operation before moving symbols

New series are often built by a short sequence of transformations: substitute, multiply, differentiate, integrate, or combine. The challenge is not any single operation but keeping coefficients, powers, centers, and convergence conditions synchronized through several steps.

A written transformation pipeline makes the reasoning auditable. Begin with a trusted parent series, label each operation, perform it on both sides, and update the interval. When only a finite number of terms is required, truncate only after the necessary operations have been completed.

**New series are built through explicit transformations**
A left-to-right pipeline shows a parent geometric series, differentiation, substitution, and multiplication, with the convergence condition carried below each stage.
New series are built through explicit transformations. An operation pipeline with function and series sides updated in parallel.

Concept

Transformation rule of thumb

Start from a named series with its domain, perform one algebraic operation, rewrite the coefficient pattern, and translate the domain before simplifying.

Guided walkthrough

From

e^u=\sum_{n=0}^{\infty}\frac{u^n}{n!},

set $u=-x^2$ and multiply by $x$ :

xe^{-x^2}=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{n!},

valid for every real $x$ .

Worked example

Differentiate after a substitution

Find a power series for

\frac{x}{(1-x^2)^2}.

Start with

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

Differentiate with respect to $u$ :

\frac1{(1-u)^2}=\sum_{n=0}^{\infty}(n+1)u^n.

Set $u=x^2$ and multiply by $x$ :

\frac{x}{(1-x^2)^2}=\sum_{n=0}^{\infty}(n+1)x^{2n+1}, \qquad |x|<1.

Common mistake

The order of operations matters

Differentiating with respect to $x$ before or after substituting $u=x^2$ produces different chain-rule factors. State which variable each derivative uses.

Interactive checku4b-building_new_series-01

Find a series for $xe^{-x^2}$ .

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Show hint

Substitute $-x^2$ into the exponential series, then multiply by $x$ .

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Exercise

Find a series for $\cos(2x)$ .

Exercise

Find a series for $x^2/(1-x)$ .

Exercise

Find a series for $1/(1+4x^2)$ .

Exercise

Track the convergence interval in every transformation.

After the explanation

Use the section idea

Reading lens

Build new expansions from a small verified library using explicit substitutions, products, derivatives, or integrals.

Mental model

A standard series is a reusable identity with a domain, not a formula fragment detached from convergence.

Decision

Name the source series, apply one transformation at a time, and transform its interval alongside it.

Common trap

Recognizing a pattern but omitting the substitution's effect on the interval.

Check yourself

Can you reconstruct the expansion and its domain without memorizing the final line?

Source & rights

Original instruction with traceable references.

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