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Infinite Series and Sequences

Samuel Baker

Samuel Baker

10 min read

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Study Guide Overview

This study guide covers series analysis, focusing on sequences and series (arithmetic, geometric, harmonic, power, alternating, Taylor, and Maclaurin). It details six convergence/divergence tests: nth Term, Limit Comparison, Direct Comparison, Integral, Alternating Series, and Ratio. Finally, it explains Lagrange and Alternating Series Error Bounds, radius/interval of convergence, and power series representation including derivatives and integrals.

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Exam Weighting

  • Only on the BC Test
  • 17-18% of the test
  • Roughly 7 to 8 multiple choice questions
  • Always one FRQ on this unit (usually FRQ 6)

Developing Understanding

Well, after going through countless limits, derivatives, and integrals, you’ve made it here! Welcome to Unit 10, the last unit of AP Calculus BC! Before we move on, give yourself a pat on the back for all your hard work!

This unit builds from past knowledge of limits, differentiation, and integration, but applies this to a new concept — series. This unit emphasizes the study of series analysis. If you haven’t ever heard of analysis in a mathematical sense before, this is just the study of functions and how they behave over time.

You will learn 6 tests for different types of series to glean information about them. It is important that you know the conditions for all the tests and what they tell you. You will also need to understand the two types of error bounds you will encounter in this unit. The concepts in this unit may seem difficult at first, but with a little practice, you will be able to understand them.

Mathematical Practices

Four of the College Board's mathematical practices for AP Calculus are used in this unit, which are outlined below.

1) Apply appropriate mathematical rules or procedures, with and without technology.

This means that you should know how to calculate the error bounds to analyze series, with the alternating error bound for alternating series and the Lagrange Error Bound for power series.

2) Explain how an approximated value relates to the actual value.

This means that you know how to interpret Lagrange and alternating series error bounds as the maximum error of an infinite series given a partial series.

3) Identify a re-expression of mathematical information presented in a given representation.

This means that you know how to rewrite any function as an infinite power series.

4) Apply an appropriate mathematical definition, theorem, or test.

There are many tests that can be used to determine series convergence and divergence. This means that it is up to you to find the one to use to fit that series.

Main Ideas for this Unit

  • nth Term Test/Limit Test
  • Limit Comparison Test
  • Direct Comparison Test
  • Integral Test
  • Alternating Series Test
  • Ratio Test
  • Lagrange E...