Transcript Sequence
Sequences
Chapter 9(1)
Sequence Definition and Notation
A Sequence is a function whose domain is the
positive integers. We use subscript notation to
denote terms of a sequence.
a1 , a2 , a3 , a4 , ...
Terms of a Sequence
There are several ways to find terms of a sequence
In terms of n
an n 3
Recursively
based on past terms
an 3an1 7
a1 1 3 4
a1 6 (must be given)
a2 2 3 5
a3 3 3 6
a2 3(6) 7 25
a3 3(25) 7 82
a20 20 3 23 a4 3(82) 7 253
Limit of a Sequence
A sequence that approaches a
value is said to Converge.
1
2
, 14 , 18 , 161 , ...
1 1
2 8
1
32
1
64
, , , ,...
Converges to 0
Converges to 0
1, , , , ... Convergesto
2
3
1
2
3
5
4
7
, 14 , 18 , 0,...
Diverges
You can’t tell if a sequence converges
just by looking at it.
1
2
Limit of a Sequence
Let L be a real number and let f be a function where
Lim f ( x) L
x
If fn = an for each positive integer n, then
Lim an L
x
This ties sequences to functions and we can
then use what we have learned about limits of
functions to find limits of sequences.
Absolute Value Theorem
If Lim an 0, then Lim an 0
n
n
This allows you to find limits when
the signs change in a sequence.
1
2
,
1
4
, 81 ,
1
16
,...
Converges to 0
Show if the sequence has
a limit and find the limit.
The sequence is:
n 2
an 2
n 1
2
11
a1 32 , a2 65 , a3 10
, ...
x 2
Lim 2
x x 1
2
1
So Lim an 1
x
Show if the sequence has
a limit and find the limit.
Lim1
Lim e
a1 2, a2 2.25, a3 2.37
The sequence is:
x
a n 1
1 n
n
1 x
x
Ln 1 1
x
x 1
Lim e
xLn 1 1x
x
L’Hopital
x
Lim e
x
So Lim an e
x
1
1 x1
e
Show if the sequence has
a limit and find the limit.
The sequence is:
a1 4, a2 4 34 , a3 4 89
Lim 5 n12
x
a n 5 n12
5n 2 1
Lim
5
2
x
n
So Lim an 5
x
Show if the sequence has
a limit and find the limit.
an
( n1)!
( n2)!
The sequence is:
a1 2!3! 13 , a2 4!3! 14 , a3 5!4! 15
Lim
( n 1)!
( n 2)!
Lim
1
n2
x
x
( n 1)( n )( n 1) ...
( n 2)( n 1) n ( n 1)...
1
0
Monotonic Sequence
A sequence with terms that are
non-decreasing
a1 a2 a3 . . . an
Or non-increasing
a1 a2 a3 . . . an
Bounded Sequence
A sequence is bounded above if
there is a real number M such that
an M for all n
A sequence is bounded below if
there is a real number N such that
N an for all n
Is the sequence monotonic
and/or bounded?
Monotonic
No Upper Bound
Lower Bound:
an
n2
n 1
a1 12 , a2 34 , a3 94 , ...
Lim an
n
1
2
(Since it is monotonic it starts
at it’s lowest value)
Graphing a sequence
an
n2
n 1
Select MODE, SEQ, and DOT
Type in the function
Under Window set the n values from 1 to 10
and the x & y max & min