Calculus II · Unit 4B · lesson

Algebra with Power Series

Add, scale, and multiply power series using coefficient logic and the Cauchy product on a shared region of convergence.

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Section overview

Algebra and calculus with power series

What this section is building

Add, scale, and multiply power series using coefficient logic and the Cauchy product on a shared region of convergence.

Notice

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

Decide

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

Avoid

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

Concept

Learning objectives

add, subtract, shift, and multiply power series while tracking coefficients and convergence intervals.

Algebra with Power Series

Explanation

Infinite algebra must be performed coefficient by coefficient

Power series can be added and subtracted like polynomials on a common interval of convergence. Multiplication uses a convolution: the coefficient of $x^n$ receives contributions from every pair of powers whose exponents add to $n$ . Writing several low-degree terms before using sigma notation keeps the index structure visible.

The resulting identity is valid at least where both original series converge absolutely. Endpoints may require fresh analysis. Algebraic manipulation can create useful representations, but it does not erase convergence conditions. An identity copied without its interval is incomplete.

Bridge

Infinite polynomials follow familiar algebra, with convergence conditions

Power series can be added, subtracted, shifted, substituted, and multiplied much like polynomials inside a common interval of convergence. Addition combines matching powers. Multiplication uses a convolution: the coefficient of $x^n$ receives contributions from every pair of exponents adding to $n$ .

The bookkeeping matters. Multiplying only matching-index terms misses most products. For approximation, one often needs only coefficients through a specified degree, so a diagonal coefficient table can prevent unnecessary expansion.

Read this graph as text

Product coefficients come from diagonals. A grid of coefficient products is organized by total exponent. Entries on the same diagonal all contribute to one power of x. A coefficient grid whose diagonals feed successive powers. Preserve the misconception control described in the lesson and storyboard.

Written labels, distinct line styles, markers, and fill patterns communicate every relationship in product coefficients come from diagonals; color is never the only cue.

Why it matters: A coefficient grid whose diagonals feed successive powers.

Product coefficients come from diagonals

A grid of coefficient products is organized by total exponent. Entries on the same diagonal all contribute to one power of x.

Product coefficients come from diagonals. A coefficient grid whose diagonals feed successive powers.

Proof idea

Diagonals collect equal total degree

\left(\sum_{j\ge0}a_jx^j\right)\left(\sum_{k\ge0}b_kx^k\right),

terms with $j+k=n$ all contribute to $x^n$ . Their sum is the convolution coefficient $c_n=\sum_{k=0}^{n}a_kb_{n-k}$ .

Concept

Cauchy product

If $A(x)=\sum a_nx^n$ and $B(x)=\sum b_nx^n$ , then formally

A(x)B(x)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}a_kb_{n-k}\right)x^n.

Guided walkthrough

Square the geometric series

Since

\frac1{1-x}=1+x+x^2+\cdots,

squaring gives

\frac1{(1-x)^2}=1+2x+3x^2+4x^3+\cdots =\sum_{n=0}^{\infty}(n+1)x^n,

valid for $|x|<1$ .

Worked example

Multiply only as far as the requested degree

Suppose

f(x)=1+x+x^2+O(x^3),\qquad g(x)=1-x+2x^2+O(x^3).

Then

f(x)g(x)=1+(1-1)x+(2-1+1)x^2+O(x^3) =1+2x^2+O(x^3).

The $x^2$ coefficient collects $1\cdot2x^2$ , $x\cdot(-x)$ , and $x^2\cdot1$ .

Common mistake

Series multiplication is not term-by-term multiplication

The coefficient of $x^n$ is $\sum_{k=0}^{n}a_kb_{n-k}$ , not simply $a_nb_n$ .

Interactive checku4b-algebra_with_power_series-01

What is the coefficient of $x^n$ in $(1+x+x^2+\cdots)^2$ ?

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

Show hint

Count pairs $(k,n-k)$ with nonnegative indices.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

Add the series for $e^x$ and $e^{-x}$ to identify even powers.

Exercise

Multiply $(1+x+x^2+\cdots)(1-x)$ .

Exercise

Find the first five terms of the product of $\sum x^n$ and $\sum(-x)^n$ .

Exercise

Explain why endpoint behavior may change after algebraic combination.

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.

Algebra with Power Series

Explanation

Infinite algebra must be performed coefficient by coefficient

Bridge

Infinite polynomials follow familiar algebra, with convergence conditions

Read this graph as text

Written labels, distinct line styles, markers, and fill patterns communicate every relationship in product coefficients come from diagonals; color is never the only cue.

Why it matters: A coefficient grid whose diagonals feed successive powers.

Product coefficients come from diagonals

A grid of coefficient products is organized by total exponent. Entries on the same diagonal all contribute to one power of x.

Product coefficients come from diagonals. A coefficient grid whose diagonals feed successive powers.

Proof idea

Diagonals collect equal total degree

\left(\sum_{j\ge0}a_jx^j\right)\left(\sum_{k\ge0}b_kx^k\right),

terms with $j+k=n$ all contribute to $x^n$ . Their sum is the convolution coefficient $c_n=\sum_{k=0}^{n}a_kb_{n-k}$ .

Concept

Cauchy product

If $A(x)=\sum a_nx^n$ and $B(x)=\sum b_nx^n$ , then formally

A(x)B(x)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}a_kb_{n-k}\right)x^n.

Guided walkthrough

Square the geometric series

Since

\frac1{1-x}=1+x+x^2+\cdots,

squaring gives

\frac1{(1-x)^2}=1+2x+3x^2+4x^3+\cdots =\sum_{n=0}^{\infty}(n+1)x^n,

valid for $|x|<1$ .

Worked example

Multiply only as far as the requested degree

Suppose

f(x)=1+x+x^2+O(x^3),\qquad g(x)=1-x+2x^2+O(x^3).

Then

f(x)g(x)=1+(1-1)x+(2-1+1)x^2+O(x^3) =1+2x^2+O(x^3).

The $x^2$ coefficient collects $1\cdot2x^2$ , $x\cdot(-x)$ , and $x^2\cdot1$ .

Common mistake

Series multiplication is not term-by-term multiplication

The coefficient of $x^n$ is $\sum_{k=0}^{n}a_kb_{n-k}$ , not simply $a_nb_n$ .

Interactive checku4b-algebra_with_power_series-01

What is the coefficient of $x^n$ in $(1+x+x^2+\cdots)^2$ ?

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

Show hint

Count pairs $(k,n-k)$ with nonnegative indices.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

Add the series for $e^x$ and $e^{-x}$ to identify even powers.

Exercise

Multiply $(1+x+x^2+\cdots)(1-x)$ .

Exercise

Find the first five terms of the product of $\sum x^n$ and $\sum(-x)^n$ .

Exercise

Explain why endpoint behavior may change after algebraic combination.

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.

Algebra with Power Series

Related resources

Original instruction with traceable references.

Algebra with Power Series

Related resources

Original instruction with traceable references.