10.1: Infinite Sequences and Summation Notaion

A sequence is a function whose domain is all natural numbers.

 represents the first time of the sequence, while  denotes a positive integer, or the nth term.
Sequences often appear as  , representing the first three terms of the sequence, as well as the nth term.

An infinite sequence is a function whose domain is the set of positive integers.

For example: Find the eighth term of the sequence,  with the first four terms: 1/3, 1/2, 3/5, 2/3...
Eighth term:
 n=8
= 4/5

There are two basic forms of writing sequences, recursive  and explicit.


Recursive Form






On the other hand, the same sequence may be represented in this way as well:

Explicit Form




Summation Notation
Summation notation is another alternative form of a sequence that can be used to find the sum of multiple terms of an infinite sequence.

Summation Notation
 

k represents the lower limit, while m represents the upper limit.
k also represents the index of summation.

This picture also re-explains summation notation in slightly different terms.


There are also two rules that apply to the sums of a constant.

Theorems on the Sum of a Constant:

1) (c being a constant)


2)


There are also a few theorems that derive from the following original theorem on sums.

Theorem on Sums


Addition

Subtraction


Constants
 for every real number c


To evaluate a sum:

Find the sum:



To find the sum, substitute the integers 1 (lower limit) through 3 (upper limit) in for k and add the resulting terms together.

(1+1) + (2+1) + (3+1)= 9

That's it for now, thank you!

Julia Wilkins