Series

Scheme of work: Year 13 A-Level Further Maths: Core Pure 2: Series

Prerequisite Knowledge

  1. Method of Differences:
  1. Arithmetic and Geometric Sequences: Understanding of common differences and ratios.
  2. Sigma (Σ) Notation: Knowing how to interpret and evaluate summations.
  3. Basic Differentiation: Understanding rates of change.
  1. Maclaurin Series:
  1. Power Series: Representing functions as an infinite sum of terms, based on powers of the variable.
  2. Factorials: Recognising factorial notation and its properties.
  3. Differentiation: Understanding the process and being able to differentiate functions multiple times.
  4. Function Evaluation: Evaluate functions and their derivatives at specific points, particularly x = 0.
  1. Series Expansion of Compound Functions:
  1. Basic Series Expansions: Familiarity with the basic expansions, e.g., for ex, sin x and cos x
  2. Function Composition: Understanding of how to compose two functions.
  3. Chain Rule: Differentiating compound functions.
  4. Integration: For understanding the connections between power series and their corresponding functions.

Success Criteria

  1. Method of Differences:
    • Understand and Apply the Technique: Students should be able to recognize when the method of differences can be applied to a given sequence or series and implement the technique appropriately.
    • Identify Patterns: Successfully identify and deduce summation patterns in sequences and series.
    • Use Sigma Notation: Properly employ and interpret sigma (Σ) notation in relevant contexts.
  2. Maclaurin Series:
    • Derive Maclaurin Expansions: Students should derive the Maclaurin series for basic functions, such as ex, sin x and cos x
    • Identify and Use the Formula: Understand and apply the general formula for the Maclaurin expansion.
    • Apply in Context: Use the Maclaurin series to approximate functions near x = 0 and recognize the radius and interval of convergence.
  3. Series Expansion of Compound Functions:
    • Recognize Basic Expansions: Know the standard expansions for basic functions.
    • Differentiate and Integrate Series: Perform calculus operations on power series.
    • Form Compound Series: Students should form series expansions of compound functions using known series expansions, especially when a new function can be expressed in terms of basic functions with known expansions.
    • Apply Chain Rule: Implement the chain rule correctly for relevant series expansion problems.

Teaching Points

  • Method of differences: The method of differences is a recursive method for finding the nth difference of a function. It can be used to find the derivative of a function or to approximate the function with a polynomial. The main teaching points for the method of differences are:
    • How to find the first few differences of a function.
    • How to use the differences to find the derivative of a function.
    • How to use the differences to approximate a function with a polynomial.
  • Maclaurin series: A Maclaurin series is a power series centred at a point. It can represent a function as an infinite sum of terms. The main teaching points for the Maclaurin series are:
    • How to construct a Maclaurin series for a function.
    • How to use a Maclaurin series to approximate a function.
    • How to find the radius of convergence of a Maclaurin series.
  • Series expansion of compound functions: The series expansion of a compound function is a way to represent a composite function as an infinite sum of terms. The main teaching points for the series expansion of compound functions are:
    • How to find the series expansion of a composite function.
    • How to use the series expansion to approximate a composite function.
    • How to find the radius of convergence of the series expansion.

In addition to these main teaching points, it is also important to emphasize the importance of convergence when discussing these topics. Students should understand that not all series converge, and that the radius of convergence of a series tells us how far from the centre the series converges.

Common Misconceptions

  1. Method of Differences:
    • Skipping Steps: Overlooking certain differences or stages in the process, leading to incorrect conclusions.
    • Sigma Errors: Misapplying or misunderstanding the sigma (ΣΣ) notation, leading to incorrect summations or formulations.
    • Assumptions: Assuming a series can be resolved using the method of differences when it’s not appropriate.
    • Arithmetic Mistakes: Simple arithmetic errors when calculating differences or summations, especially under exam pressure.
  2. Maclaurin Series:
    • Truncation Issues: Truncating the series too early or late, leading to significant approximation errors.
    • Formula Recall: Misrecalling the general formula for Maclaurin series.
    • Differentiation Errors: Making mistakes in the repeated differentiation process required for the Maclaurin coefficients.
    • Convergence Oversight: Not considering or stating the interval of convergence when required.
    • Remainder Neglect: Ignoring or incorrectly estimating the remainder term when using a truncated series for approximations.
  3. Series Expansion of Compound Functions:
    • Misapplication: Incorrectly applying known series expansions directly to compound functions without appropriate adjustments.
    • Convergence Assumptions: Neglecting to check or incorrectly determining the new domain of convergence for the expanded compound function.
    • Integration/Differentiation Issues: Incorrectly integrating or differentiating term-by-term without proper justification or making calculation errors.
    • Mishandling Coefficients: Failing to adjust coefficients appropriately when combining series or deriving series for compound functions.

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