Pre – Calculus
Chapter 10
Section 10.1 Notes: Sequences, Series, and Sigma
Objectives:
- Investigate different types of sequences
- Use sigma notation to represent and calculate terms of a series
DEFINITIONS
Sequence: an ordered list of numbers
Term: each number in the sequence
Finite Sequence: contains a finite number of terms (has an end)
Infinite Sequence: contains an infinite number of terms (never ends)
PATTERNS: Find the next term in the pattern.
4 5 6
, , , _____,...
7 8 9
a) 1, 4, 9, 16, 25, ______, …
b) 5, 25, 125, ______, …
c)
d) 0, 3, 8, 15, 24, ______, …
e) 1, 2, 5, 10, 17, ______, …
f) 8, 16, 32, 64, ______
NOTATION
a1 represents the first term in a sequence
a5 represents the fifth term in a sequence
a100 represents the 100th term!
an represents the nth term in a sequence
FORMULA
Infinitely many sequences exist with the same first few terms. To define a unique sequence, a formula for the nth term must be
given.
This is an explicit formula that gives the nth term, an ,as a function of n.
Example: y as a function of x  y = 2x + 3
Example: an as a function of n an = n + 2
Example 1: Find the next four terms in the sequence.
a) 3, –1, –5, –9, ….
b) 18, 15, 10, 3, ….
c) Find the first four terms of the sequence given by an = n3 + 1.
CONVERGENT VS. DIVERGENT
Recall end behavior of the graph of a function from chapter 1
 As some domains (x – values) approach , some ranges approach a specific limit (y-value)
As a function, an infinite sequence may also have a limit.
 To converge: if a sequence has a limit
 To diverge: if a sequence does not have a limit
Example 4: Which examples are convergent?
1 𝑛
a) 𝑎𝑛 = −4 (− )
4
b) 𝑎𝑛 =
(−2)𝑛
c) an = -3n + 12
2
Part 2
DEFINITIONS
Series: the SUM of the terms in a sequence
Finite Series: the indicated sum of all the terms in the finite sequence.
Infinite Series: the indicated sum of all the terms in the infinite sequence.
nth partial sum: the sum of the first n terms.
 Denoted by Sn
 Determine all terms from a1 to an, then find the sum of all these terms.
Example 5:
a) Find the fifth partial sum of an = n2 – 3.
b) Find S4 of 𝑎𝑛 =
6
2𝑛
INFINITE SERIES
Since an infinite series does not have a finite number of terms, you might assume that the infinite series has no sum.
However, some infinite series do have sums. In order for an infinite series to have a fixed sum, S, the infinite sequence associated with
the series must converge to 0.
Compare these sequences: which will have a sum?
 1, 4, 7, 10, ….
 0.1, 0.01, 0.001, ….
SIGMA NOTATION

Also known as “Summation Notation”
The lower bound describes which term to begin with.
The upper bound describes which term to end with. [If the upper bound is , the sigma notation represents an infinite sum]
Add all terms between the lower and upper bound.
Example 6: Write the notation in expanded form. Then find the sum.
a)
b)