Topic: Sequences and Series

Subtopic: Introduction to Sequences and Series

**Overview**

Pop quiz! Can you complete these sequences?

a. 5, 11, 17, 23, 29, ___, ___, ___

b. 2, 4, 8, 16, 32, ___, ___, ___

c. 3, 5, 7, 11, 13, ___, ___, ___

d. 1, 1, 2, 3, 5, 8, ___, ___, ___

Think about them before reading on.

Answers:

a. 35, 41, 47. To get the subsequent terms you add 6. This is an example of an __arithmetic sequence__.

b. 64, 128, 256. To get the subsequent terms you multiply by 2. This is an example of a __geometric sequence__.

c. 17, 19, 23. This is simply a sequence of prime numbers.

d. 13, 21, 34. Can you see that the subsequent terms are formed by combining previous terms (1+1=2, 1+2=3, 2+3=5, 3+5=8, etc.)? This is an example of a __recursive sequence__.

This lesson introduces general sequences (lists) and series (sums) -- the notation, terminology, and processes. We'll follow this lesson with a concentration on specific types of sequences and series (arithmetic and geometric). This material will be used particularly in Calculus III when we will write functions such as e^{x} as a sum of infinitely many rational terms (`e^x = 1 + x + x^2/(2!) + x^3/(3!) + x^4/(4!) + ...`). This will enable us to perform calculus on rational expressions rather than on the (more complicated) transcendental function itself.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 8.1.1 Evaluate or simplify a factorial algebraically and electronically
- 8.1.2 Write the first few terms of a sequence given the a
_{n}term - 8.1.3 Write the next few terms of a sequence given the first few terms
- 8.1.4 Understand alternating sequences and recursive sequences
- 8.1.5 Know the difference between sequences and series
- 8.1.6 Evaluate or simplify a series algebraically and electronically
- 8.1.7 Use proper mathematical notation and format when working with factorials, sequences, and series
- 8.1.8 Apply properties of series as needed

**Terminology**

Define: sequence, finite sequence, infinite sequence, series, finite series, infinite series, terms (of a sequence or series), summation, summation notation, sigma notation, upper limit, lower limit, index of summation, sum, partial sum, sequence of partial sums, factorial, alternating sequence, Fibonacci sequence, recursively-defined (a.k.a. recursive) sequence, properties of series

**Text Notes**

The Fibonacci Sequence is the most famous example of a recursive sequence. If you haven't investigated this very rich topic before or would like to learn more, please explore the links below.

If your text discusses electronically producing "dot" graphs of a sequence you may SKIP these examples/problems throughout the chapter.

**Supplemental Resources** (optional)

Pages to explore the Fibonacci sequence and its connections to Pascal's Triangle and the Golden Mean:

Biography of Leonardo Fibonacci

Dr. Knott's Fibonacci Numbers and Nature ~ a must read!

Pascal's Triangle and its Patterns

The Golden Ratio/Section/Mean/Number

Want more sites to explore or some books to read? Just ask! This is a favourite topic of mine, practically a hobby :)