How to Use This Practice Set
Taylor and Maclaurin series problems are much easier when you recognize the structure of the problem before doing algebra. The key skills are not just memorizing formulas, but knowing when to start from a familiar power series, when to differentiate or integrate a series, and when a remainder estimate is required.
For each problem, first identify what the question is really asking: find a polynomial, find a series, determine convergence, approximate a value, or bound the error. Then choose the shortest valid method.
Essential Series to Know
The following Maclaurin series are especially important for AP Calculus BC. You should be able to recognize and manipulate these without rebuilding them from the definition every time.
Practice Problems
Find the fourth-degree Maclaurin polynomial for .
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Find the third-degree Taylor polynomial for centered at .
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Write the Maclaurin series for .
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Find the Maclaurin series for .
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Find the Maclaurin series for .
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Express as a power series centered at 0 and state its interval of convergence.
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Determine the radius and interval of convergence of .
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Determine the interval of convergence of .
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Use a Maclaurin polynomial to approximate .
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Use the Lagrange error bound to estimate the maximum error when the fourth-degree Maclaurin polynomial for is used to approximate .
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Find the first four nonzero terms of the Taylor series for centered at .
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Use a known power series to find a power series for .
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Use a known power series to find a power series for .
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Determine the radius of convergence of .
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Suppose the Maclaurin series for is known. Explain how you could obtain a power series for without calculating derivatives from scratch.
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A function is represented by the power series . Find the value of the function at , assuming the series converges there.
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Which familiar Maclaurin series can be used to represent ? Write the resulting series.
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Determine the interval of convergence for .
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Use the first three nonzero terms of a suitable Maclaurin series to approximate .
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Find the first four nonzero terms of the Maclaurin series for , and state the radius of convergence.
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Worked Strategies
If the function resembles a familiar series, do not immediately compute derivatives. Start with the known series and manipulate it.
Once you know a power series, you can often create another one by differentiating or integrating term by term inside its interval of convergence.
The ratio or root test usually determines the radius of convergence, but it does not automatically determine what happens at the endpoints. Always test each endpoint separately.
A fourth-degree Taylor polynomial contains powers through . It may have fewer than five nonzero terms if some derivatives vanish.
A Maclaurin series is centered at . A Taylor series centered at another value uses powers of . Make sure the center matches the question.
AP Calculus BC Checklist
- Know the Maclaurin series for , , , , and .
- Be able to build Taylor polynomials from derivatives.
- Be able to substitute expressions into known power series.
- Know how differentiation and integration affect a power series.
- Use the ratio or root test to determine the radius of convergence.
- Test endpoints separately to determine the complete interval of convergence.
- Understand how Taylor polynomials approximate functions near their center.
- Use the Lagrange remainder theorem when an explicit error bound is requested.
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