Representing Functions as Power Series

Derive function representations by valid series operations, fixing constants and rechecking endpoints.

College Board Topic 10.15 · CED p. 195 (PDF p. 200) · CED effective Fall 2020; project course outline 2026–27.

A known series is a building block. You can substitute, multiply by a polynomial, add series on a common convergence interval, differentiate term by term, or integrate term by term. These operations are valid inside the relevant positive radius. State the interval and recheck any candidate endpoints after an operation.

Substitution into the geometric series gives $1 / (1 + x^{2}) = {\sum_{n = 0}}^{\infty} (- 1)^{n} x^{2 n}$ for |x|<1. Integrating from 0 to x yields $a r c \tan x = {\sum_{n = 0}}^{\infty} (- 1)^{n} x^{2 n + 1} / (2 n + 1)$ . The integration constant is fixed by arctan0=0. At x=±1 the integrated series converges by alternating magnitudes; it represents arctan there as well, with a remainder tending to zero.

Differentiating $1 / (1 - x) = {\sum_{n = 0}}^{\infty} x^{n}$ gives $1 / (1 - x)^{2} = {\sum_{n = 1}}^{\infty} n x^{n - 1} = {\sum_{n = 0}}^{\infty} (n + 1) x^{n}$ on |x|<1. Reindexing changes the displayed index, not the terms. At ±1 the new terms fail to tend to zero, so those endpoints are excluded.

For the arctangent identity, a finite geometric remainder makes the endpoint justification concrete: integrating the difference after N terms gives a remainder with magnitude at most $| x |^{2 N + 1} / (2 N + 1)$ when |x|≤1. It tends to zero even at the endpoints. A convergent endpoint alone is not always enough to assert a new function identity without such reasoning.

Use the retained series-evaluation lesson and Taylor evaluation practice for further application. Show the original identity and the operation before the final answer; that makes the source of every coefficient reviewable.

Let's work through it

Start from $1 / (1 + x^{2}) = \sum (- 1)^{n} x^{2 n}$ for |x|<1. Why: Substitute −x² into the geometric identity.
Integrate from 0 to x: $a r c \tan x = \sum (- 1)^{n} x^{2 n + 1} / (2 n + 1)$ . Why: Termwise integration is valid inside the radius and both sides vanish at zero.
At ±1, test the alternating odd-reciprocal series; it converges. The integrated remainder also tends to zero. Why: Endpoint convergence and vanishing remainder justify the representation on the closed interval −1≤x≤1.

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Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.

Question 1 of 20 self-assessed

Question 1

What series follows by differentiating

{\sum_{n = 0}}^{\infty} x^{n}

inside |x|<1?

Your answer and reasoning

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Question 1: prompt and worked reasoning

What series follows by differentiating

{\sum_{n = 0}}^{\infty} x^{n}

inside |x|<1?

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

Differentiate each power; the constant term disappears.

Question 2: prompt and worked reasoning

When you integrate a known series to identify a function, how do you fix the constant?

Use an initial function value or a definite integral.

Indefinite termwise integration alone leaves a constant undetermined.

See every recall question with its answer

These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.

Retrieval 1

What series follows by differentiating

{\sum_{n = 0}}^{\infty} x^{n}

inside |x|<1?

Worked answer and reason

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

Differentiate each power; the constant term disappears.

Retrieval 2

When you integrate a known series to identify a function, how do you fix the constant?

Worked answer and reason

Use an initial function value or a definite integral.

Indefinite termwise integration alone leaves a constant undetermined.

Catch the tempting mistake

A derived series automatically keeps all original endpoints.: The radius stays the same under differentiation/integration, but endpoint behavior can change. Test the new terms at each endpoint.

Calculator workspace and limits

For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.

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Put it to work

Unit 10 problem 26 · No calculator · analytic
Unit 10 problem 28 · No calculator · analytic
Unit 10 problem 29 · No calculator · analytic
Unit 10 problem 30 · Graphing calculator required · multiple

For the teacher

Have pairs exchange a starting series and an operation, then audit coefficients, constants, radius and endpoints.

Unit planning and formative guidance →

Glossary and deeper reading · 4 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Infinite series
An infinite series is the sum of the terms of an infinite sequence, defined through its partial sums. The series converges when those partial sums approach a finite value. For example, 1/2+1/4+1/8+⋯ converges to 1. Terms approaching 0 are necessary for convergence, but that condition alone is insufficient.
Read more: 10.1 · Infinite series →
Polynomial
A polynomial in one variable is a finite sum of constant multiples of nonnegative integer powers of that variable. The degree is the highest exponent with a nonzero coefficient. For example, 2x³−x+5 is a polynomial of degree 3. Expressions containing 1/x or √x are not polynomials in x.
Geometric series
A geometric series adds terms that have a fixed common ratio. With first term a and ratio r, it converges to a/(1−r) when |r|<1. For example, 1+1/2+1/4+⋯ sums to 2. For a nonzero first term and |r|≥1, the infinite series diverges.
Read more: 10.2 · Geometric series →
Constant of integration
A constant of integration represents the arbitrary constant added when finding a family of antiderivatives. It appears because differentiating a constant gives zero. For example, every function x²+C has derivative 2x. An initial condition may determine a particular value of C.
Read more: 6.8 · Constant of integration →

Let's work through it

Try it from memory

Try it before you peek.

Question 1

Retrieval 1

Retrieval 2

Catch the tempting mistake

Put it to work

Infinite series

Polynomial

Geometric series

Constant of integration

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