≥Better Grades
Menu

Calculus II · Unit 4B · lesson

Maclaurin Series

Concept

Learning objectives

construct a Maclaurin series formally and distinguish a Taylor series from the function it may represent.

Maclaurin Series

Explanation

An infinite Taylor polynomial is a candidate representation

Letting the degree grow without bound produces the formal Maclaurin series ∑f(n)(0)xn/n!\sum f^{(n)}(0)x^n/n!. The series is built from the function, but it need not equal the function automatically. Equality requires the remainder to approach zero.

This distinction separates algebraic construction from analytic justification. Some smooth functions have all derivatives and still are not represented by their Taylor series away from the center. In ordinary calculus examples such as exponential and trigonometric functions, remainder estimates establish equality on large domains.

Bridge

A Maclaurin series records every derivative at zero

A Maclaurin series is a Taylor series centered at 00. Its coefficients form a derivative fingerprint:

f(x)∼∑n=0∞f(n)(0)n!xn.f(x)\sim \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n.

For functions with repeating derivative patterns, this produces memorable series efficiently.

The symbol “equals” requires justification. Derivative matching constructs the formal candidate series, but the function equals that infinite series only where the Taylor remainder tends to zero. This distinction separates calculating coefficients from proving representation.

Derivative cycles generate coefficient patterns. Derivative cycles for sine, cosine, and exponential functions.
Read this graph as text

Derivative cycles generate coefficient patterns. A four-step cycle shows cosine, negative sine, negative cosine, and sine, with their values at zero determining alternating even-power coefficients. Derivative cycles for sine, cosine, and exponential functions.

Written labels, distinct line styles, markers, and fill patterns communicate every relationship in derivative cycles generate coefficient patterns; color is never the only cue.

Why it matters: Derivative cycles for sine, cosine, and exponential functions.

Derivative cycles generate coefficient patterns

A four-step cycle shows cosine, negative sine, negative cosine, and sine, with their values at zero determining alternating even-power coefficients.

Derivative cycles generate coefficient patterns. Derivative cycles for sine, cosine, and exponential functions.

Proof idea

Coefficient construction and function representation are separate theorems

The derivative data uniquely determine the Taylor coefficients. Taylor's Theorem later shows that

f(x)=Tn(x)+Rn(x).f(x)=T_n(x)+R_n(x).

Only when Rn(x)→0R_n(x)\to0 does the infinite series actually equal f(x)f(x).

Concept

Maclaurin series

The Maclaurin series of ff is

∑n=0∞f(n)(0)n!xn.\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n.
Guided walkthrough

The sine pattern

The derivatives of sin⁡x\sin x cycle:

sin⁡x, cos⁡x, −sin⁡x, −cos⁡x,…\sin x,\ \cos x,\ -\sin x,\ -\cos x,\ldots

At zero the nonzero values occur at odd orders with alternating signs, giving

sin⁡x=x−x33!+x55!−⋯ .\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots.
Worked example

The cosine derivative cycle chooses the even powers

The derivatives of cos⁡x\cos x cycle:

cos⁡x,−sin⁡x,−cos⁡x,sin⁡x,…\cos x,-\sin x,-\cos x,\sin x,\ldots

At x=0x=0, their values are 1,0,−1,0,…1,0,-1,0,\ldots. Therefore

cos⁡x=1−x22!+x44!−x66!+⋯ .\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots.

Odd powers vanish because every odd derivative is zero at the center.

Common mistake

Derivative matching creates a candidate, not automatically an identity

A smooth function can have all derivatives at zero and still fail to equal its Taylor series away from zero. Equality requires a remainder argument.

Interactive checku4b-maclaurin_series-01

Write the first three nonzero Maclaurin terms of sin⁡x\sin x.

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

Show hint

Use odd powers and alternating signs.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

Write the first four nonzero terms of cos⁡x\cos x.

Exercise

Find the Maclaurin coefficients of 1/(1−x)1/(1-x).

Exercise

Explain why a formal Taylor series may fail to equal its source function.

Exercise

Identify the parity pattern of sine and cosine series.

After the explanation

Use the section idea

Reading lens

Match value and derivatives at one center, then separate the polynomial approximation from the infinite-series convergence claim.

Mental model

Taylor coefficients encode local derivative data as a polynomial of increasing degree.

Decision

Choose the center, compute the derivative pattern, divide by factorials, and state whether you need a polynomial or an infinite series.

Common trap

Assuming every smooth-looking function equals its Taylor series everywhere.

Check yourself

Can you verify the first coefficients directly from derivatives at the center?

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.