Step by step
The question
Question: find the Taylor series of 1/x at x = 1.
f(x) = 1x
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 1.
f(1) = 1
Take the derivative again and again.
f′(x) = −1x2
f′(1) = −1
f″(x) = 2x3
f″(1) = 2
f‴(x) = −6x4
f‴(1) = −6
f⁗(x) = 24x5
f⁗(1) = 24
2 Divide by n factorial
Divide each value by n factorial: 0!
= 1, 1!
= 1, 2!
= 2, 3!
= 6, 4!
= 24.
numbers : 10!, −11!, 22! = 1, −63! = −1, 244! = 1
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.
The dots mean it goes on for ever.
f(x) = 1 − (x − 1) + (x − 1)2 − (x − 1)3 + (x − 1)4 + ...
4 Check
Let's check close to 1, at x = 1.1.
Our terms give about 0.9091.
A calculator gives 0.90909 for 1 over x.
Almost the same.
Right!
x = 1.1: the terms ≈ 0.9091, 1/x ≈ 0.90909
Let's look back.
First: the derivatives.
Then: divide by n factorial.
Then: write the series.
Last: check.
Answer
f(x) = 1 − (x − 1) + (x − 1)2 − (x − 1)3 + (x − 1)4 + ...