AP Calculus BC · Unit 10: Infinite Sequences and Series
10.15 Representing Functions as Power Series
In short
This topic focuses on representing functions as power series by manipulating known series (such as the Maclaurin series for sin(x), cos(x), e^x, or 1/(1+x)). You are expected to derive new power series using operations including term-by-term differentiation, term-by-term integration, algebraic processes, substitutions, or the properties of geometric series.
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What else do you need to know about Representing Functions as Power Series?
This is a BC-only topic. The emphasis here is on algebraic manipulation and calculus operations on existing series rather than deriving series from scratch using the general Taylor polynomial formula.
Can you do these?
- I can derive a power series for a given function by applying operations like term-by-term differentiation, integration, algebraic processes, or substitutions to a known series.
How is Representing Functions as Power Series tested on the exam?
On the AP Exam, operating on a known series to create another series requires proficient, well-presented algebra. You frequently struggle with communicating your understanding of series; therefore, you must use correct notation and language to present solutions, explain reasoning, and justify conclusions. Questions often assess Skill 3.D (Apply an appropriate mathematical definition, theorem, or test). Free-response questions may require you to perform multiple manipulations on a basic series (e.g., differentiate or integrate) and subsequently determine the new interval of convergence.
Free practice questions with full explanations, and instant marking of your free-response answers.
Representing Functions as Power Series: common questions
What do you need to know about Representing Functions as Power Series for AP Calculus BC?
This topic focuses on representing functions as power series by manipulating known series (such as the Maclaurin series for sin(x), cos(x), e^x, or 1/(1+x)). You are expected to derive new power series using operations including term-by-term differentiation, term-by-term integration, algebraic processes, substitutions, or the properties of geometric series.
How is Representing Functions as Power Series tested on the AP Calculus BC exam?
On the AP Exam, operating on a known series to create another series requires proficient, well-presented algebra. You frequently struggle with communicating your understanding of series; therefore, you must use correct notation and language to present solutions, explain reasoning, and justify conclusions.
How do I know I have mastered Representing Functions as Power Series?
Check you can do each of these: derive a power series for a given function by applying operations like term-by-term differentiation, integration, algebraic processes, or substitutions to a known series.
Where does Representing Functions as Power Series fit in AP Calculus BC?
Representing Functions as Power Series is topic 10.15 in Unit 10: Infinite Sequences and Series of AP Calculus BC.
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