What a Maclaurin Series Is
A Maclaurin series is a Taylor series centered at . It represents a function as an infinite polynomial whose coefficients are determined by the function's derivatives at zero.
The first few terms are usually enough to recognize the pattern:
On AP Calculus BC, you are not expected to memorize every possible power series. Instead, you should know several standard Maclaurin series and be able to manipulate them using substitution, multiplication, differentiation, integration, and algebra.
The Most Important Maclaurin Series
The Exponential Function
The Maclaurin series for is one of the most important series in the entire course.
This series converges for every real number .
Sine
The sine function has an alternating series containing only odd powers.
Notice the pattern of signs: positive, negative, positive, negative.
Cosine
The cosine series contains only even powers.
The cosine series also alternates signs, but the powers are even rather than odd.
The Geometric Series
The geometric series is especially useful because many other series can be created from it.
For the Maclaurin series centered at zero, the most common form is:
This converges when .
The Natural Logarithm
The series for is another standard series worth recognizing.
The interval of convergence is .
A Standard-Series Cheat Sheet
These formulas are worth knowing cold. If you see one of these functions—or a simple modification of one—you should immediately think about whether a standard series can save you from computing derivatives one at a time.
How to Use Substitution
The most powerful idea is that you can replace in a known series with another expression. This lets you create a Maclaurin series for functions that are not on the standard list.
Find the Maclaurin series for .
Find the first four nonzero terms of the Maclaurin series for .
Find the Maclaurin series for .
Using Differentiation and Integration
Once you know one power series, you can often generate another by differentiating or integrating term-by-term. This is a major AP Calculus BC skill.
Use the geometric series to find a power series for .
Maclaurin Polynomial vs. Maclaurin Series
A Maclaurin polynomial contains only a finite number of terms. A Maclaurin series continues indefinitely.
The polynomial above is the degree-5 Maclaurin polynomial for . The corresponding series continues with higher-order terms.
On the AP exam, pay close attention to whether a problem asks for a polynomial, a certain number of nonzero terms, or the full series.
Common Mistakes
The coefficients in many standard series contain factorials. For example, the exponential series uses , not simply powers of . Dropping the factorial changes every term after the first two.
Sine contains odd powers, while cosine contains even powers. A quick memory trick is that starts with an odd power and starts with an even power.
If you replace with , substitute it everywhere—including inside the powers and factorial terms when appropriate. Do not replace only the first occurrence.
A power series may only represent its function on a particular interval. For example, the geometric series requires . Always check convergence when the problem asks for an interval or radius of convergence.
Practice Problems
Write the Maclaurin series for .
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Find the first four nonzero terms of the Maclaurin series for .
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Find the Maclaurin series for .
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Use a known Maclaurin series to find the first four nonzero terms of the series for .
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