Language of Sequences & Series

Language of Sequences & Series

Language of sequences & series

What is a progression?

• A progression (also called 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 progression are often called terms

• The terms of a progression are often referred to by letters with a subscript

  • This will often be the letter u

  • So in the progression above, u₁ = 1, u₂ = 3, u₃ = 5 and so on

• Each term in a progression 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 progression

  • 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

Sigma Notation

Sigma notation

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

  • For example $\sum _{k = 0}^4(2k+1)$  or  $\sum _{k = 7}^{14}(2k-13)$

  • r and k are commonly used variables within sigma notation