Chapter 13 _ Introduction to sequences and series

Chapter 13 _ Introduction to sequences and series -...

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Unformatted text preview: Chapter
13
–
Introduction
to
sequences
and
series

(13.3)
–
Extra
Credit
 
 A
numerical
sequence
is
an
ordered
list
of
numbers,
a
collection
of
items.


 
 Example:

 {1,3,5,7,9,.......... } 
 
 The
individual
item
in
a
numerical
sequence
is
called
terms
of
the
sequence.
 
 If
a
sequence
has
finite
number
of
terms,
it
is
called
a
finite
sequence.
 
 ex:

 {2,4 ,6,8 } 
 
 Similarly,
if
a
sequence
has
infinite
number
of
terms,
it
is
called
an
infinite
 sequence.
 
 ex:

 {1,3,5,..........2n + 1,..... } 
 
 
 n Ex:

Consider
the
sequence
 {a1 ,a2 ,a3 ,...........an ,..... } in which a n = 2 

 n +1 
 Write
the
first
five
terms.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Graph:
 
 
 
 
 
 
 
 A
recursive
sequence
is
defined
by

 
 1 b1 = 0 bn = ( bn −1 + 36)
 2 
 This
kind
of
sequence
is
defined
in
terms
of
its
previous
term.
 
 Find
the
next
four
terms
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Definition:

A
sequence
is
a
function
whose
domain
is
either
the
set
of
natural
 numbers
or
the
set
of
non‐negative
numbers
 
 
 How
do
you
find
the
sums
of
terms
of
a
sequence?
 
 Ex:

Given
a
sequence
defined
by
 ak = ( 3k _ 2)2 .

Find
the
sum
of
the
first
five
terms.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Written
in
short
hand:
 ∑( 3k − 2)2 
 
 k =1 n ∑ 
 
 = sigma noatation defining adding of consecutive terms = ∑ a k = a1 + a2 + a3 + .........an k =1 n Ex:

Find
 ∑ ln( k =1 8 k +1 )
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Rewrite
using
the
sigma
notation


 
 x x 2 x3 x9 + + + ................. + 
 1! 2! 3! 9! 
 
 
 
 
 
 
 
 
 
 
 Arithmetic
Sequence
and
Series

(13.4)
 
 
 Definition:

An
arithmetic
sequence
is
of
the
form
a,
a+d,
a+2d,
a+3d,
……….
 where
a
is
the
first
term
and
d
is
the
difference
between
the
consecutive
terms.
 
 The
nth
term
of
an
arithmetic
sequence
can
be
written
as:
 
 an = a + ( n − 1)d 
 
 Ex:
Determine
the
50th
term
of
the
arithmetic
sequence
defined
by
3,
7,
11,
15,
19,…..
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex:

Determine
the
arithmetic
sequence
whose
3rd
term
is
11
and
the
10th
term
is
‐3
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Formula:

Sum
of
an
Arithmetic
series:
 
 n( a + an ) 
 Sn = 2 
 Ex:

Find
the
sum
of
the
first
40
terms
of
a
sequence
defined
by
{2,
5,
8,
11,
14,
17,….}
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Useful
sums
to
recognize
for
Calculus

(pg
999
in
text)
 
 n n( n + 1) 1)∑ k = 2 k =1 2) ∑k k =1 n 3 n 2 = n( n + 1)( 2n + 1) 6 2 
 ⎛ n( n + 1)⎞ 3)∑ k = ⎜ ⎟ ⎝ 2⎠ k =1 
 
 
 
 
 
 
 
 Geometric
sequences
and
series
(13.5)
 
 Definition:

A
Geometric
sequence
or
progression
is
of
the
form

 
 a,
ar,
ar2,ar3,ar4,………..,arn‐1,………….


 
 where
a
and
r
are
non
zero
elements
 
 
 Note:

1)

The
nth
term
of
a
geometric
sequence
is
 an = ar n −1
 
 2)

The
ratio
of
succeeding
terms
of
a
geometric
sequence
is
constant,
that
is
 
 an +1 ar ( n +1) −1 rn = = n −1 = r 
 an ar n −1 r 
 Ex:

Find
the
11th
term
of
the
geometric
sequence
12,
6,3,
3/2,
¾,
………
 
 
 
 
 
 
 
 
 
 
 
 
 A
finite
Geometric
series
is
of
the
form
a+ar+ar2+ar3+ar4+………..+arn‐1
 
 a(1 − r n ) The
Sum
 Sn = 
 1−r 
 Ex:

Evaluate
the
sum
of
the
finite
Geometric
series

 
 333 3 3 + + + + ........................... + 11 
 248 2 
 
 
 
 
 
 
 An
infinite
Geometric
series
is
of
the
form
a
+
ar
+
ar2+
ar3+……………
 
 If
 r < 1,
the
sum
of
such
a
series
can
be
found
and
is
given
by
the
following
formula:
 
 a 










where
a
is
the
1st
term
and
r
is
the
ratio
between
the
consecutive
 S= 1−r terms.
 
 
 Ex:

Find
a
fraction
equivalent
to

0.333333333333333
 
 
 
 
 
 ...
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