Topics

Sequences and series

The topic is divided into the following subtopics:
  • Introduction to sequences
  • Arithmetic sequences
  • Geometric sequences
  • Limit of a sequence
  • Infinite series

Introduction to sequences

Part I:
  • Ways to specify a sequence
  • Finding of one or more members of a sequence
Part II:
  • Defining a sequence (nth term formula or recurrent relation)
  • Properties of sequences (strictly increasing or strictly decreasing, nonincreasing, nondecreasing, upper or lower bounded, bounded)
  • Finding the nth term of a sequence
Part III:
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Arithmetic sequences

Part I:
  • Defining a sequence (nth term formula or recurrent relation)
  • Finding the nth term of a sequence
  • Finding the common difference of a sequence
Part II:
  • Finding the nth term of a sequence – complex problems
  • Finding the common difference of a sequence – complex problems
  • Sum of the first n terms of a sequence
  • Systems of equations containing sequence terms
Part III:
  • Word problems
  • Equations and inequalities containing sums of sequences

Geometric sequences

Part I:
  • Defining a sequence (nth term formula or recurrent relation)
  • Finding the nth term of a sequence
  • Finding the common ratio of a sequence
Part II:
  • Finding the nth term of a sequence – complex problems
  • Finding the common ratio of a sequence – complex problems
  • Sum of the first n terms of a sequence
  • Systems of equations containing sequence terms
Part III:
  • Word problems
  • Combinations of arithmetic and geometric sequences

Limit of a sequence

Part I:
  • Evaluation of limits of sequences containing polynomials and rational expressions
  • Limit laws – sum of limits, difference of limits, product of limits and ratio of limits laws
Part II:
  • Evaluation of limits with trigonometric, exponential and logarithmic functions
Part III:
  • Use of the limit of the sequence (1+1/n)^n
  • Evaluation of limits with radicals
  • Evaluation of limits containing sums of sequences

Infinite series

Part I:
  • Summation notation
  • Finding of the first term and of the common ratio of a geometric sequence
  • The sum of an infinite geometric series
Part II:
  • Periodic numbers (repeating decimals)
  • Finding of all x for which a series diverges or converges
  • Solving equations with infinite series
  • Word problems
Part III:
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