Calculus II · Unit 4B · lesson

Building New Series from Known Ones

Construct new power series through substitution, multiplication, differentiation, and integration while transforming the domain correctly.

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Section overview

Standard series and new expansions

What this section is building

Construct new power series through substitution, multiplication, differentiation, and integration while transforming the domain correctly.

Notice

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

Decide

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

Avoid

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

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 $xe^{-x^2}$ .

Your work stays on this device. No account or AI grader is used.

Show hint

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

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

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.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Building New Series from Known Ones

Explanation

Transformation is usually faster than repeated derivatives

Bridge

Plan the operation before moving symbols

Read this graph as text

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. An operation pipeline with function and series sides updated in parallel. Preserve the misconception control described in the lesson and storyboard.

Written labels, distinct line styles, markers, and fill patterns communicate every relationship in new series are built through explicit transformations; color is never the only cue.

Why it matters: An operation pipeline with function and series sides updated in parallel.

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 $xe^{-x^2}$ .

Your work stays on this device. No account or AI grader is used.

Show hint

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

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

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.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Building New Series from Known Ones

Related resources

Original instruction with traceable references.

Building New Series from Known Ones

Related resources

Original instruction with traceable references.