10.12 Maclaurin series
Some functions can be written as an "infinite power series", which is a sum in the form
How would we find the values of for a specific function?
Differentiation! Let’s take :
|(By differentiating both sides)|
|(By differentiating both sides again)|
|(By differentiating both sides once more)|
How do we find the value of a coefficient? Another algebra trick - plug in a specific value for , in this case . This is because every term in the power series, except the constant one, depends on and thus is zero when is zero. In order to find the value of the next constant, we can just differentiate, which brings all the powers down by one.
This leads directly to the general case (i.e. the value of )
Where means the value of the nth derivative at the point . Therefore, overall, we can write the power series of any function1818 18 Note that are some additional conditions - the function must be infinitely differentiable (we can keep differentiating forever) and each derivative must be defined at the point . as
Maclaurin series of common functions
This is just a list, which is also given in the formula booklet.
10.12.1 Derivation of the sum of an infinite geometric series
Prove the formula for a sum of an infinite geometric series using Maclaurin series.
We know1919 19 If not, see Section 3.4.2 that
This can be derived by finding an expression for the th derivative of .
Finding the first few derivatives,
By the chain rule, this gives that the first derivative is equal to
The second derivative is equal to
The third is
And so on (this can be proved by induction). Therefore, the Maclaurin expansion for the function is
10.12.2 A lower bound for the factorial function
This is a really cool method (which I discovered originally on Terence Tao’s blog) to find a lower bound for the factorial function2020 20 The factorial function is defined as . The first step is to consider the Maclaurin series for , i.e.
If we specify that , then all the terms in the series are positive, so any individual term of the series will be smaller than the term we have selected and thus we can select the term where () which satisfies
This is closely related to Stirling’s formula (a closed-form approximation for the factorials which shows up in quite a few places) - the linked blog post gives the full derivation.