Calculus II · Unit 4B · lesson
Standard Maclaurin Series
Recognize and use the standard exponential, sine, cosine, geometric, logarithm, and arctangent series with their convergence conditions.
Section overview
Constructing Taylor and Maclaurin seriesWhat this section is building
Recognize and use the standard exponential, sine, cosine, geometric, logarithm, and arctangent series with their convergence conditions.
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
recall and use the standard series for exponential, sine, cosine, and geometric functions.
Standard Maclaurin Series
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 .
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.
Read this graph as text
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. Aligned pattern rows for exponential, sine, and cosine series. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in standard maclaurin series have visible coefficient patterns; color is never the only cue.
Why it matters: Aligned pattern rows for exponential, sine, and cosine series.
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.
Standard library
Using four terms,
The next terms are already small because factorials grow rapidly.
Substitute into a standard series without losing the pattern
From
set . Then
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 .
Transform the entire term, not only the first power
When replacing by , every occurrence becomes . Writing loses powers of two.
u4b-standard_maclaurin_series-01Write the first three nonzero Maclaurin terms of .
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Use even powers with alternating signs.
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Write five terms of .
Use the sine series to approximate .
Explain why cosine contains only even powers.
Compare coefficient decay in geometric and exponential series.
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