Derivation of Taylor Series Approximation
Description:
The Taylor Series Approximation is a way of describing any differential function as a sum of polynomials
Idea. Let’s say we have a function and we want to plug in some variable into it but it ends up we don’t
know how to calculate that function at a specific point well let’s say we know how to calculate the
function at the point a which just happens to be close to the point x which we want to calculate it at.
And let’s say were are able to calculate the functions successive derivatives at point a. A great example
of this property would be the trig functions such as sine & cosine. We know that taking the derivative of
either will give us the other, we also know what sine & cosine are easily calculated for certain values. Tis
means with Taylor Series we can convert sine and cosine into a sum of polynomials which can be
calculated easily at any point. This is actually how calculators do it.
Mathematics needed:
o
o
Fundamental Theorem of Calculus (FTC)
Integration by Parts
Derivation
We will start with the FTC
𝑥
∫ 𝑓(𝑡) 𝑑𝑡 = 𝑓(𝑥) − 𝑓(𝑎)
𝑎
We find that we can write it like using Integration by parts on the integral
𝑥
′
𝑓(𝑥) = 𝑓(𝑎) + ∫ 𝑓(𝑡)
𝑑𝑡
𝑎
=
𝑥
′
𝑓(𝑎) + [𝑓(𝑡)
𝑡|
𝑎
𝑥
′′
− ∫ 𝑓(𝑡)
𝑡𝑑𝑡 ]
𝑎
𝑥
′
′
′′
= 𝑓(𝑎) + 𝑓(𝑥)
𝑥 − 𝑓(𝑎)
𝑎 − ∫ 𝑓(𝑡)
𝑡𝑑𝑡
𝑎
𝑥
'
′
′′
With FTC we can re-write f(x)
=𝑓(𝑎)
+ ∫𝑎 𝑓(𝑡)
𝑑𝑡 and get
𝑥
𝑥
′
′′
′′
(𝑥 − 𝑎) + 𝑥 ∫ 𝑓(𝑡)
= 𝑓(𝑎) + 𝑓(𝑎)
𝑑𝑡 − ∫ 𝑓(𝑡)
𝑡𝑑𝑡
𝑎
𝑎
𝑥
′ (𝑥
′′
= 𝑓(𝑎) + 𝑓(𝑎)
− 𝑎) + ∫ 𝑓(𝑡)
(𝑥 − 𝑡)𝑑𝑡
𝑎
Using Integration by parts again and we get
𝑥
=
′ (𝑥
𝑓(𝑎) + 𝑓(𝑎)
− 𝑎) +
′′
𝑓(𝑡)
𝑥
𝑡2
𝑡2
′′′
(𝑥𝑡 − )| − ∫ 𝑓(𝑡)
(𝑥𝑡 − ) 𝑑𝑡
2! 𝑎
2!
𝑎
′ (𝑥
= 𝑓(𝑎) + 𝑓(𝑎)
− 𝑎) +
𝑥
𝑥 2 ′′
𝑎 2 ′′
𝑡2
′′
′′′
𝑓(𝑥) − 𝑎𝑥𝑓(𝑎)
+ 𝑓(𝑎)
− ∫ 𝑓(𝑡)
(𝑥𝑡 − ) 𝑑𝑡
2
2
2!
𝑎
We will once again use the same strategy to get rid of the function of x and we get
= 𝑓(𝑎) +
′ (𝑥
𝑓(𝑎)
𝑥
𝑥 2 ′′
𝑎 2 ′′
𝑥 2 𝑥 ′′′
𝑡2
′′
′′′
− 𝑎) + 𝑓(𝑎) − 𝑎𝑥𝑓(𝑎) + 𝑓(𝑎) − ∫ 𝑓(𝑡) 𝑑𝑡 − ∫ 𝑓(𝑡)
(𝑥𝑡 − ) 𝑑𝑡
2
2
2 𝑎
2!
𝑎
′ (𝑥
= 𝑓(𝑎) + 𝑓(𝑎)
− 𝑎) +
𝑥
2
𝑥 2 ′′
𝑎 2 ′′
𝑥2
′′
′′′ 𝑡
𝑓(𝑎) 𝑎𝑥𝑓(𝑎)
+ 𝑓(𝑎)
+ ∫ 𝑓(𝑡)
( − 𝑥𝑡 − ) 𝑑𝑡
2
2
2!
2!
𝑎
′ (𝑥
= 𝑓(𝑎) + 𝑓(𝑎)
− 𝑎) +
𝑥
2
1 ′′
𝑥2
′′′ 𝑡
𝑓(𝑎) (𝑥 − 𝑎)2 + ∫ 𝑓(𝑡)
( − 𝑥𝑡 − ) 𝑑𝑡
2!
2!
2!
𝑎
From here we would do integration by parts again and we would find that there is a pattern to this
process. In the end we are able express it as an infinite sum.
∞
𝑓(𝑥)
𝑖 (𝑥
𝑓(𝑎)
− 𝑎)𝑖
=∑
𝑖!
𝑖=0