Calculus II · Unit 4B · lesson

Standard Maclaurin Series

Concept

Learning objectives

recall and use the standard series for exponential, sine, cosine, and geometric functions.

Standard Maclaurin Series

Explanation

A small library prevents repeated derivative work

Several series occur so frequently that they should become as familiar as derivative rules. The geometric, exponential, sine, and cosine series form a basic library. Their coefficient patterns reflect structural properties: factorial decay for exponential and trigonometric functions, parity for sine and cosine, and constant coefficients for the geometric series.

Memorization should be supported by reconstruction. If a sign or factorial is forgotten, derivative patterns recover the series. Each identity also includes a convergence domain. The exponential, sine, and cosine series converge for every real input; the geometric series requires $|x|<1$ .

Bridge

A small library prevents repeated reinvention

Several Maclaurin series appear so often that they function like a table of derivatives or antiderivatives. The exponential series uses every power with factorial denominators; sine uses alternating odd powers; cosine uses alternating even powers. Their patterns come directly from derivative cycles.

Memorization is useful only when paired with structure. A student should know the center, the first several terms, the general term, and the interval of convergence. From there, substitution and scaling can generate many new series without differentiating a complicated composition repeatedly.

**Standard Maclaurin series have visible coefficient patterns**
Aligned rows show e to the x using all powers, sine using alternating odd powers, and cosine using alternating even powers.
Standard Maclaurin series have visible coefficient patterns. Aligned pattern rows for exponential, sine, and cosine series.

Concept

Standard library

\frac1{1-x}=\sum_{n=0}^{\infty}x^n,\quad |x|<1,

e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!},\quad \sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!},

\cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}.

Guided walkthrough

Using four terms,

e^{0.1}\approx1+0.1+\frac{0.1^2}{2}+\frac{0.1^3}{6}=1.105166\ldots

The next terms are already small because factorials grow rapidly.

Worked example

Substitute into a standard series without losing the pattern

From

\sin u=u-\frac{u^3}{3!}+\frac{u^5}{5!}-\cdots,

set $u=2x$ . Then

\sin(2x)=2x-\frac{(2x)^3}{3!}+\frac{(2x)^5}{5!}-\cdots =2x-\frac{4}{3}x^3+\frac{4}{15}x^5-\cdots.

The powers remain odd and the signs still alternate. Because the sine series converges for every real input, the transformed series also converges for every real $x$ .

Common mistake

Transform the entire term, not only the first power

When replacing $u$ by $2x$ , every occurrence becomes $(2x)^{2n+1}$ . Writing $2x^{2n+1}$ loses powers of two.

Interactive checku4b-standard_maclaurin_series-01

Write the first three nonzero Maclaurin terms of $\cos x$ .

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

Show hint

Use even powers with alternating signs.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

Write five terms of $e^{-x}$ .

Exercise

Use the sine series to approximate $\sin(0.2)$ .

Exercise

Explain why cosine contains only even powers.

Exercise

Compare coefficient decay in geometric and exponential 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.