2 2
= ’’()
′′() 2!
2
=
Since 2=2! Differentiating and repeating the above process infinitely many times, we have:
3() 3!
4() 4!
3
=
4
=
…
Since all derivatives of ( ) are identical to those of 𝑃( ) ; the curves change in an identical way and, since they start from the same point, the curves are identical, i.e.
() 0!
′()( − ) 1!
′′()( − )2 2!
( ) ≡ 𝑃( ) ≡
+
+
…
Condensing this infinite sum into sigma notation yields the formula given in the introduction.
Examples:
Below are examples of the Taylor series centred at the point p=0 for common functions. Taylor series centred at the point p=0 are also known as Maclaurin series.
2 2!
3 3!
1!
𝑥 = 1 +
+
+
. . . . .
2 2!
4 4!
6 6!
( ) = 1 −
+
−
. . . .
3 3!
5 5!
7 7!
𝑖( ) = −
+
−
+ ⋯
Placing any value for x into the right hand side polynomial gives an approximation for the actual function on the left hand side of the equation. For example, putting = 0 into the trigonometric functions gives the standard results cos(0) = 1 and 𝑖(0) = 0 .
The series also works if composite functions are used. For example,
(2 ) 2 2!
(2 ) 3 3!
2 1!
2𝑥 = 1 +
+
+
…
Some properties of the imaginary number 𝑖 include:
𝑖 1 = 𝑖 , 𝑖 2 = −1 , 𝑖 3 = −𝑖 , 𝑖 4 = 1 …
Generalising to all integers:
𝑖 4+1 = 𝑖 , 𝑖 4+2 = −1 , 𝑖 4+3 = −𝑖 , 𝑖 4 = 1
where n is a non-negative integer.
Replacing with 𝑖 in the exponent of 𝑥 gives:
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