SCHOLAR Study Guide: Sequences and Series

SCHOLAR Study Guide Advanced Higher Mathematics: Algebra, Proof, and Number Theory - Sequences and Series

1. Introduction

• Authored by: Fiona Withey, Karen Withey

• Reviewed by: Margaret Ferguson

• Originally by: Jane S Paterson, Dorothy A Watson

• Institution: Heriot-Watt University, Edinburgh, UK

• First published: 2019, Updated: 2022

• Copyright © 2022 SCHOLAR Forum

2. Acknowledgements

• Thanks to those who contributed to the SCHOLAR programme including education authorities, colleges, teachers, and students.
     - Permission to use material granted by SQA for past papers assessments.
     - Financial support acknowledged from the Scottish Government.

3. Topic Overview: Sequences and Series

3.1 Learning Objectives

• Identify geometric and arithmetic sequences.

• Determine linear recurrence relations and calculate terms.

• Identify limits of convergent sequences and solve recurrence relations.

• Define various types of sequences and understand convergence.

4. Topics in Detail

4.1 Looking Back: Pre-requisites from Advanced Higher

• Conditions for Differentiability:
    - A function is continuous if
    $ext{lim}<br>olimits_{h o 0} f(x + h) = f(a)$.
    - A function is differentiable at a point if it possesses a tangent at that point.
    - Higher Derivatives: Denoted by either $f^{(n)}(x)$ or $rac{d^n y}{dx^n}$.

• Product Rule:
    $k(x) = f(x) g(x)$ then
    $k^{ ext{'}}(x) = f^{ ext{'}}(x) g(x) + f(x) g^{ ext{'}}(x)$.

• Quotient Rule:
    $k(x) = rac{f(x)}{g(x)}$, then
    $k^{ ext{'}}(x) = rac{f^{ ext{'}}(x)g(x) - f(x)g^{ ext{'}}(x)}{[g(x)]^2}$.

• Derivative information for trig functions:
    - $ext{sec}(x) = rac{1}{ ext{cos}(x)}$
    - $ext{cosec}(x) = rac{1}{ ext{sin}(x)}$
    - $ext{cot}(x) = rac{1}{ ext{tan}(x)}$

4.2 Recurrence Relations

• Definition of a Sequence: A sequence is a list of numbers or terms with a specific pattern defined by a rule for the nth term.

• Simple Recurrence Relations:
    - Example: Sequence {2, 5, 8, 11,…}
      - Rule: Start with 2 and add 3 each time.
      - General formula: $u_n = 3n - 1$.
    - Another Example: Fibonacci sequence {1, 1, 2, 3, 5, 8,…}
      - Rule: Each term is the sum of the two preceding terms.
      - General formula is complex but involves the golden ratio.

4.3 Types of Sequences

• Arithmetic Sequence: Defined as a pattern where each term is generated by adding a constant (the common difference d) to the previous term, expressed as:
    - $u_{n+1} = u_n + d$.

• Geometric Sequence: A sequence where each term is found by multiplying the previous term by a constant (the common ratio r), expressed as:
    - $u_{n+1} = ru_n$.

4.4 Convergence and Limits

• Definition of Convergence: A sequence converges if it approaches a specific limit as n approaches infinity.
    - A convergent sequence tends toward a limit L.
    - Null sequences converge to 0.
    - Bounded sequences remain within limits.

4.5 Sums to Infinity

• Convergent Geometric Series: A geometric series is said to converge if the absolute value of the common ratio r is less than 1:
    - $S_{ ext{∞}} = rac{a}{1 - r}$, where a is the first term.

4.6 Maclaurin Series

• Maclaurin's Theorem: A function that can be expressed as a power series about 0:
    - $f(x) = f(0) + f'(0)x + rac{f''(0)}{2!}x^2 + rac{f'''(0)}{3!}x^3 + …$

• Each function such as $e^x$, $sin(x)$, $cos(x)$ can be expressed in this form.

4.7 Examples of Deriving Series

• For example, to find the Maclaurin series expansion for $e^{-x}$, we differentiate:
    - $f(x) = e^{-x}, f'(x) = -e^{-x}, f''(x) = e^{-x}$ and evaluate at x=0.

4.8 Exercises and Applications

• Students are encouraged to apply concepts by solving exercises relating to sequences, recurrence relations, limits, and series sums.

4.9 Testing Understanding

• Comprehensive end-of-topic tests covering all aspects of sequences and series, focusing on applying learned concepts to various problems and scenarios.