Sequences/SeriesBOMLA LACYMATH

SUMMER 2015

Overview

* In this unit, we’ll be introduced to some very unique ways of displaying patterns of numbers known as sequences and series. This allows us to count large sets of numbers more easily. Infinite sequences, especially those with finite limits, are involved in some key concepts of Calculus. Infinite series are at the heart of integral Calculus.

Sequences A sequence is an ordered pattern of numbers.

Sometimes abbreviated as or .

Example

Sequences

Some sequences are finite (stop at a certain value), and some sequences are infinite (continue forever).

Finite Sequences

Some sequences are finite (stop at a certain value), and some sequences are infinite (continue forever).

Examples

Infinite Sequences

Some sequences are finite (stop at a certain value), and some sequences are infinite (continue forever).

Examples

Infinite Sequences

Some sequences are finite (stop at a certain value), and some sequences are infinite (continue forever).

If the limit of an infinite sequence approaches a certain value, then it converges. If the limit of an infinite sequence is infinite or nonexistent, then it diverges.

Arithmetic vs. Geometric

Arithmetic Sequence – Terms are related by a common difference.

Geometric Sequence – Terms are related by a common ratio.

Arithmetic Sequences The common difference can be found by subtracting two consecutive terms.

Examples

Arithmetic Sequences In an infinite arithmetic sequence, there is a formula to tell us the exact term anywhere in the sequence.

Geometric Sequences The common ratio can be found by dividing two consecutive terms.

Examples

Geometric Sequences In an infinite geometric sequence, there is a formula to tell us the exact term anywhere in the sequence.

Formulas for Sequences

Recursive vs. Explicit

Series A series is the sum of the terms in a sequence.

This can be written in summation notation, where we use the Greek letter Sigma.

There are formulas for both arithmetic and geometric series.

Sum of an Arithmetic Series Our formula is…

Sum of a Finite Geometric Series Our formula is…

Sum of an Infinite Geometric Series

You can only use this if !!!

Our formula is…