Algebra/Chapter 15/Introduction to Sequences and Series

Sequence

Within algebra, a sequence is an ordered list of numbers, called terms. It is often described by an equation or rule. The domain often begins with $n=1$, but doesn't have to. Sequences can be separated into two types, finite and infinite.

Finite sequence

A finite sequence is one that has a list of terms in a certain order with a beginning and ending term.

ex: {1, 2, 3, 4, 5}

Infinite sequence

An infinite sequence is a sequence with an indefinite list of terms, this happens when the sequence is extended to all positive integers.

ex: {1, 2, 3, 4, 5,...}

Writing sequences

When writing sequences there are two formats that are most commonly used, function notation and subscript notation. Function notation is modeled as $a(n)$, subscript notation on the other hand is modeled as $a_{n}$. In both of these notations $n$ denotes the number in the sequence. Both of these notations will elicit the same results.

Series

A series is the expression formed by adding the terms in a sequence. Much like sequences, series can be split into finite and infinite. A finite series is when the terms of a finite sequence are added together, an infinite series expresses the sum of the terms in an infinite sequence and has an infinite number of addends. This is represented with summation notation (sigma notation)

Summation notation

$\sum _{k=1}^{n}k$

This is an example of a finite series, breaking it down it can be seen as three separate parts that come together to make up the summation notation. at the top of the $\sum$ (sigma) is the upper limit, the upper limit is where to stop adding terms. At the bottom of the sigma is the lower limit or the starting point where a variable from m to n is equal to our first term. In front of the sigma is a summation term or what is being added together.