Unit 6: Sequences and Series

Into Into Sequences

Sequence

A Sequence is a discrete structure to represent an ordered list
ex)
- -4,-1,2,5, … (... means the series is infinite)
- 27,9,3,1 …
Sequences can be organised into similar families of sequences.
Families made because of:
- Common features
- Characteristics that make each different family different
Two types of functions
- Recursive
  - To get the next term you need the one before it
- Explicit
  - Linear or exponential
  - Links the term number to the value

Summation Notation and Sigma

Sigma Notation

The Greek capital letter sigma, Σ, (equivalent to the capital “S” in English) is used in mathematics as a compact way to indicate sums. We use sigma notation (summation notation) as a shorter way to write a long list of numbers or terms being added together.

read as “The sum of k squared from k=1 to 3” is equivalent to 14
Examples:

Arithmetic Series

An arithmetic sequence is a sequence with an addition(or subtraction) generator. The number added to each term to get the next term is called the common difference.

Breaking it down further
- When a sequence can be generated by adding a constant to each previous term, it is called an arithmetic sequence.
You can compute the sum of a arithmetic series from using the formula or understanding the reversing and adding trick. The Gauss method shows us that by writing the sequence in reverse order and paring off the terms with the original sequence we can add up the terms easily by knowing the number of terms in the sequence.
General formula:

Example:

Geometric Series

A geometric sequence is a sequence with a multiplication (or division) generator. The number multiplied by each term to get the next term is called the common ration or the multiplier.

Breaking it down further
- When a sequence can be generated by multiplying constant times each previous term, it is called geometric sequence.
A geometric series works similar to a arithmetic but different, a geometric starts with the a term and then you multiply r to every next term till its ar^(n-1). The new series ends in ar^n instead of ar^(n-1) and then you subtract the two series and get ar^n-a or the formula
General Formula:

example:

Infinite Geometric and Harmonic Series

An infinite geometric series is the sum of an infinite geometric sequence
- The series has no final term
- Formula - a1 + a1r + a1r^2 + a1r^3 ...
  - a1 - first term
  - r - common ratio
In an inifnite series where common ratio is greater than 1, the terms will continue to get larger and there will be no final answer
If the common ratio is between -1 and 1, there can be a sum
- Formula for this scenario - S = a1/1-r
When an infite series has a sum, it is called a converegent series
- Sum of this series, Sn, is called the partial sum
Sigma notation can be used to represent an ifinite series
Ex)

Limits and Induction

Limits:

A limit tells us as a number gets infinitely larger or smaller what number it approaches. We know that in polynomials the largest exponent Is the dominant term when there are competing numbers. For non-polynomial functions you determine the dominant term in the numerator and denominator to evaluate the limit. You plug in a very large or very small number to evaluate the limit.
A limit exists when there is a vertical asymptote or when the denominator is factored and set equal to 0. A horizontal asymptote when the degree of the num is small than the denom. Slant asymptote when the degree of the un is larger than the denominator. There is no limit when there is a hole.

Induction:

Induction is a particular method of proving certain types of mathematical statements of formulas, particularly those involving positive integers. For those particular mathematical statements, induction provides a very effective method of proof.

Example:

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