Calculus II · Unit 4B · lesson

Algebra with Power Series

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.

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 $(1+x+x^2+\cdots)^2$ ?

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Show hint

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

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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.

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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