Taylor Series and Eulerβs identity
Andrew Ng
The square root of -1 is denoted by the letter π . π is the ratio of the circumference of a circle to its diameter, and e is defined as the constant satisfying the following limit:
1
lim ββ
(1 +
)
The latter two can be shown to be both irrational (i.e. they canβt be written in the form where and are integers) and transcendental, i.e. they arenβt the solution of a polynomial equation with rational coefficients, e.g. x-1=0. What would go through your mind if I told you that:
π = β1
Mind-blowing eh? From this π can be shown to equal β π 2 It seems completely inconceivable to the uninitiated that raising an infinitely long constant to another infinitely long constant multiplied by an imaginary number gives such a normal integer as -1.
It all arises from the following formula:
β
() ( ) !
( β )
( ) = β
=0
where () ( ) denotes the th derivative of ( ) evaluated at the point .
Intuitive Proof:
Given any function ( ) that is defined and infinitely differentiable over [ββ, β] , we can approximate it as a polynomial which we call the power series of ( ) . Letβs call it π( ) . It must then be equal to ( ) at at least one point, . Then, let:
( β ) 2 + β―
π( ) = 0
+ 1
( β ) + 2
Evaluating at , () = 0
. Therefore:
( β ) 2 + β―
π( ) = () + 1
( β ) + 2
Differentiating π( ) :
( β ) 2
πβ( ) = 1
+ 2
( β ) + 3
Repeating the above process, we have
1
= β() .
Differentiating Pβ(x) and repeating the above process, we have
21
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