Step by step
The question
Question: find the Maclaurin series of 1/(1 − x).
The Maclaurin series: the nth derivative at 0, divided by n factorial, times x to the n.
Maclaurin means a = 0.
1 The derivatives
Then put in x = 0: f(0) is 1.
Take the derivative again and again.
2 Divide by n factorial
Divide each value by n factorial: 0!
= 1, 1!
= 1, 2!
= 2, 3!
= 6, 4!
= 24.
Careful!
Do not forget to divide by n factorial.
The x squared term is f″ over 2 factorial.
3 Write the series
Put them together: each number times x to its power.
The dots mean it goes on for ever.
4 The pattern
The pattern: x to the n, every number 1.
With Σ the whole series fits on one line.
5 Check
Let's check close to 0, at x = 0.1.
Our terms give about 1.1111.
A calculator gives 1.11111 for 1/(1 − x).
Almost the same.
Right!
Let's look back.
First: the derivatives.
Then: divide by n factorial.
Then: write the series.
Then: the pattern.
Last: check.