IB DP Maths: AI HL复习笔记1.3.1 Language of Sequences & Series

What is a sequence?

• A sequence is an ordered set of numbers with a rule for finding all of the numbers in the sequence
  • For example 1, 3, 5, 7, 9, … is a sequence with the rule ‘start at one and add two to each number’
• The numbers in a sequence are often called terms
• The terms of a sequence are often referred to by letters with a subscript
  • In IB this will be the letter u
  • So in the sequence above, u₁ = 1, u₂ = 3, u₃ = 5 and so on
• Each term in a sequence can be found by substituting the term number into formula for the n^th term

What is a series?

• You get a series by summing up the terms in a sequence
  • E.g. For the sequence 1, 3, 5, 7, … the associated series is 1 + 3 + 5 + 7 +  …
• We use the notation S_n to refer to the sum of the first n terms in the series
  • S_n = u₁ + u₂ + u₃ + … + u_n
  • So for the series above S₅ = 1 + 3 + 5 + 7 + 9 = 25

What are increasing, decreasing and periodic sequences?

• A sequence is increasing if u_n+1 > u_n for all positive integers
  • i.e. every term is greater than the term before it
• A sequence is decreasing if u_n+1 < u_n for all positive integers
  • i.e. every term is less than the term before it
• A sequence is periodic if the terms repeat in a cycle

What is sigma notation?

• Sigma notation is used to show the sum of a certain number of terms in a sequence
• The symbol Σ is the capital Greek letter sigma
• Σ stands for ‘sum’
  • The expression to the right of the Σ tells you what is being summed, and the limits above and below tell you which terms you are summing

• Be careful, the limits don’t have to start with 1