1 Appendix E: Sigma Notation

2 Definition: Sequence A sequence is a function a(n) (written a n ) who’s domain is the set of natural numbers {1, 2, 3, 4, 5, ….}. a n is called the general term of the sequence. The output of a sequence can be written as {a 1, a 2, a 3, …, a n-1, a n, a n+1, …}, where a n is a term in a sequence, a n-1 is the term before it, and a n+1 is the term after it. Sequences can be either finite (their domains are {1, 2, 3, …, n}) or infinite (their domains are { 1, 2, 3, ….} ). A sequence who’s input for the next term in the sequence is the value of the previous term is called a recursive sequence.

3 Definition: Arithmetic Sequence An arithmetic sequence is a sequence generated by adding a real number (called the common difference, d) to the previous term to get the next term. The general term of an arithmetic is given by a n = a 1 + d(n – 1) where a 1 and d are any real numbers. Example Find the general term of the 7/3, 8/3, 3, 10/3, ….

4 Definition: Geometric Sequence A geometric sequence is a sequence generated by multiplying the previous term by a real number (called the common ratio r). The general term of a geometric sequence is given by a n = a 1 r (n – 1) where a 1 and r are any real numbers, is called an geometric sequence. Example Find the general term sequence 2, 2/5, 2/25, 2/125, … TI: seq(a x, x, i start, i stop)

5 Definition: Series A finite series is the sum of a finite number of terms of a sequence. An infinite series is the sum of an infinite number of terms of a sequence. We use sigma notation to denote a series. The series does not have to start at i = 1, but i must be in the domain of a i.

6 Definition: Geometric Sequence The n th partial sum is the sum of the first n terms of a sequence. It MUST start at i = 1 with partial sum notation. An infinite sum is the sum of all the terms of an infinite sequence.

7 Definition: Example TI: sum(seq(a x, x, i start, i stop))

8 Definition: Example

9 Definition: Series For a finite arithmetic series, For an infinite arithmetic series, For a finite geometric series, For an infinite geometric series, if | r | < 1. It DNE otherwise.

10 Definition: Example

11 Definition: Series Formulas Let c be a constant and n a positive integer.

12 Definition: Series Formulas 9. Write a formula for the series in terms of n: 10. If the interval [a, b] is split into n equal subintervals, write a sequence x i that represents the x coordinate of the left side, midpoint, and right side of each subinterval. 11. Show that