Download - 11.1 An Introduction to Sequences & Series
11.1 An Introduction to Sequences & Series
p. 651
• What is a sequence?• What is the difference between
finite and infinite?
Sequence:• A list of ordered numbers separated by
commas. • Each number in the list is called a term.• For Example:
Sequence 1 Sequence 2 2,4,6,8,10 2,4,6,8,10,…
Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5Domain – relative position of each term (1,2,3,4,5)
Usually begins with position 1 unless otherwise stated.
Range – the actual terms of the sequence (2,4,6,8,10)
Sequence 1 Sequence 2 2,4,6,8,10 2,4,6,8,10,…
A sequence can be finite or infinite.
The sequence has a last term or final
term.(such as seq. 1)
The sequence continues without
stopping.(such as seq. 2)
Both sequences have a general rule: an = 2n where n is the term # and an is the nth term.
The general rule can also be written in function notation: f(n) = 2n
Examples:• Write the first 6
terms of an=5-n.• a1=5-1=4• a2=5-2=3• a3=5-3=2• a4=5-4=1• a5=5-5=0• a6=5-6=-1
• 4,3,2,1,0,-1
• Write the first 6 terms of an=2n.
• a1=21=2• a2=22=4• a3=23=8• a4=24=16• a5=25=32• a6=26=64
• 2,4,8,16,32,64
Examples: Write a rule for the nth term.
The seq. can be written as:
Or, an=2/(5n)
• The seq. can be written as:
2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,…
Or, an=2n+1
,...625
2,125
2,252,
52 .a
,...52,
52,
52,
52
4321
,...9,7,5,3 .b
Explicit Formula
• When the rules for a sequence are written so that the nth term can be calculated immediately, then it is expressed as an explicit formula.
Examples:an=2/(5n) or , an=2n+1
Example: write an EXPLICIT rule for the nth term.
• 2,6,12,20,…
• Can be written as:1(2), 2(3), 3(4), 4(5),…
Or, an=n(n+1)
Recursive Form• When given one or more of the initial terms and
then defining the terms that follow using those previous terms is called a recursive formula.
Example: find the 4th term if 1 11, 2 1n na a a n
12 2 2 12a a
2 1 3 4a 13 3 2 13a a
23 5 4 5 9a a
4 4 1 342 1 7 16a a a
Try These …
Find the sixth term of the sequences.
1 18, 2 7n na a a 1 13, ( 2)n na a a
-96 39
A Famous Sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, …
Found in flower petals, tree branches, bones in the human hand …
THE FIBONACCI SEQUENCE
Convergent – Divergent Sequences
If a sequence approaches a constant as the value of n gets large, the sequence is said to converge.
If a sequence does NOT converge, it is divergent.
Divergent or Convergent?
na
13
26
39
412
3na n
This sequence does not approach a constant … DIVERGENT
Divergent or Convergent?
Hint: for explicit formulas, graph on your calculator and evaluate what happens when x gets large.
For recursive formula, find the first several terms of the sequence and determine what will happen.
Divergent or Convergent?
Try these …642na n
1 19, 4n na a a
3( 1)nna