Transcript Special Sequences and Series

12.7 (Chapter 9) Special
Sequences & Series
Fibonacci Sequence:
1, 1, 3, 5, 8, 13, …




Describes many patterns of numbers found in
nature.
a1 = 1 and a2 = 1
How do we arrive at the next term?
It was used to investigate the reproductive
habits of rabbits in ideal conditions in 1202.

An important series used to define the irrational
number e, developed by Leonhard Euler. It can be
expressed as the sum of the following infinite series:
1 1 1 1
1
e  1      ... 
1! 2! 3! 4!
n!
The binomial theorem can be used to derive the series for e. Let
k be any positive integer and apply the binomial theorem to:
 1
 1  k (k  1)  1  k ( k  1)( k  2)  1 
1    1  k   
  
  
2!  k 
3!
 k
k
k
k
2
3
k (k  1)(k  2)...1  1 
...+
  
k!
k
k
 1   1  2 
 1  2  1
11   11  1  
1 1   1   ...
k   k  k 
k  k  k


11

 ... 
2!
3!
k!
Then find the limit as k increases without bound.
k
1 1 1
 1
lim 1    1  1     ...
k 
2! 3! 4!
 k
Thus e can be defined as:
k
1 1 1
 1
e = lim 1   or e = 1  1     ...
k 
2! 3! 4!
 k
The value of ex can be approximated using the following series
known as the exponential series.

n
2
3
4
x
x
x
x
e x    1  x     ...
2! 3! 4!
n 0 n !
Ex 1
Use the first five terms of the
exponential series and a
calculator to approximate the
0.65
value of e to the nearest
hundredth.
Trigonometric Series
2n
2
4
6
8

1
x


x
x
x
x
cos x  
 1      ...
2! 4! 6! 8!
 2n  !
n 0

n
2 n 1
3
5
7
9

1
x


x x x
x
sin x  
 x      ...
3! 5! 7! 9!
n  0  2n  1 !

n

The two trig series are convergent for all
values of x. By replacing x with any angle
measure expressed in radians and carrying
out the computations, approximate values of
the trig functions can be found to any desired
degree of accuracy.
Ex 2
Use the first five terms
of the trig series to find

the value of sin 3
Euler’s Formula
Derived by replacing x by i in the exponential series, where i is an
imaginary # and  is the measure of an angle in radians.
2
3
4
(
i

)
(
i

)
(
i

)
ei  1  i 


 ...
2!
3!
4!
e  1  i 
i
2
i
3

4
 ...
2!
3! 4!
Group the terms according to whether they contain i.
2
4
6
3
5
7










i
e  1 


 ...   i   


 ... 
2! 4! 6!
3! 5! 7!

 

The real part is exactly cos and the imaginary part is exactly sin .
Therefore:
Euler's Formula:
ei  cos   i sin 
Can be used to write a complex number, a + bi,
in its exponential form, rei .
a  bi  r (cos   i sin  )
=re
i
Ex 3
Write in exponential form:
2 i 2
1 i 3

Recall: There is no real number that is the
logarithm of a negative number. You can use
a special case of Euler’s Formula to find a
complex number that is the natural logarithm
of a negative number.
ei  cos   i sin 
ei  cos   i sin  (let  = )
ei  1  i (0)
ei  1 (so, ei  1  0)
Take natural logo f both sides:
ln ei  ln( 1)
i  ln( 1)
The natural log of a negative # -5, for k>0, can be
defined using ln(-k) = ln(-1)k or ln(-1) + ln k.
Ex 4
Evaluate:

ln(-540)

ln(-270)