Step by step
The question
Question: find the Taylor polynomial of degree 4 of ln x at x = 1.
f(x) = ln x
The Taylor series: the nth derivative at the point a, divided by n factorial, times x minus a to the n.
f(x) = Σ f⁽ⁿ⁾(a)n!(x − a)n
1 The derivatives
Then put in x = 1: f(1) is 0.
f(1) = 0
Take the derivative again and again.
f′(x) = 1x
f′(1) = 1
f″(x) = −1x2
f″(1) = −1
f‴(x) = 2x3
f‴(1) = 2
f⁗(x) = −6x4
f⁗(1) = −6
2 Divide by n factorial
Divide each value by n factorial: 0!
= 1, 1!
= 1, 2!
= 2, 3!
= 6, 4!
= 24.
numbers : 00!, 11!, −12! = −12, 23! = 13, −64! = −14
Careful!
Do not forget to divide by n factorial.
The x squared term is f″ over 2 factorial.
Careful: not the (x − 1)² term = f″(1) × (x − 1)² ✗
3 Write the series
Put them together: each number times (x − 1) to its power.
Stop at the power 4.
P₄(x) = (x − 1) − (x − 1)22 + (x − 1)33 − (x − 1)44
4 Check
Let's check close to 1, at x = 1.1.
Our terms give about 0.09531.
A calculator gives 0.09531 for ln x.
Almost the same.
Right!
x = 1.1: the terms ≈ 0.09531, ln x ≈ 0.09531
Let's look back.
First: the derivatives.
Then: divide by n factorial.
Then: write the series.
Last: check.
Answer
P₄(x) = (x − 1) − (x − 1)22 + (x − 1)33 − (x − 1)44