Calculus II · Unit 4B · lesson
Maclaurin Series
Construct Maclaurin series from derivative patterns and distinguish a formal coefficient series from a proven function representation.
Section overview
Constructing Taylor and Maclaurin seriesWhat this section is building
Construct Maclaurin series from derivative patterns and distinguish a formal coefficient series from a proven function representation.
Taylor coefficients encode local derivative data as a polynomial of increasing degree.
Choose the center, compute the derivative pattern, divide by factorials, and state whether you need a polynomial or an infinite series.
Assuming every smooth-looking function equals its Taylor series everywhere.
Learning objectives
construct a Maclaurin series formally and distinguish a Taylor series from the function it may represent.
Maclaurin Series
An infinite Taylor polynomial is a candidate representation
Letting the degree grow without bound produces the formal Maclaurin series . 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.
A Maclaurin series records every derivative at zero
A Maclaurin series is a Taylor series centered at . Its coefficients form a derivative fingerprint:
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.
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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. Preserve the misconception control described in the lesson and storyboard.
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.
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.
Coefficient construction and function representation are separate theorems
The derivative data uniquely determine the Taylor coefficients. Taylor's Theorem later shows that
Only when does the infinite series actually equal .
Maclaurin series
The Maclaurin series of is
The sine pattern
The derivatives of cycle:
At zero the nonzero values occur at odd orders with alternating signs, giving
The cosine derivative cycle chooses the even powers
The derivatives of cycle:
At , their values are . Therefore
Odd powers vanish because every odd derivative is zero at the center.
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.
u4b-maclaurin_series-01Write the first three nonzero Maclaurin terms of .
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Use odd powers and alternating signs.
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Write the first four nonzero terms of .
Find the Maclaurin coefficients of .
Explain why a formal Taylor series may fail to equal its source function.
Identify the parity pattern of sine and cosine series.
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