Series expansions of compound functions

Compound functions are those that are made up of two or more simpler functions. For example, f(g(x)) or h(tan(x)).
In Core Pure Mathematics 2, we look at how we can expand these compound functions into power series, a useful tool in many areas of applied and pure mathematics.

The Maclaurin series:

The Maclaurin series is a type of Taylor series that is expanded about the point x = 0. It is a useful tool for approximating functions near this point.
The formula for the nth term of a Maclaurin series is ((f^(n)(0))/n!) * x^n.
Certain functions have well-known Maclaurin series that you should memorise for Core Pure Mathematics 2. For example, the Maclaurin series for e^x, sin(x), cos(x), ln(1+x) and (1+x)^n.

Taylor Series:

The Taylor series is a more general form of the Maclaurin series that can be expanded about any point a, not just 0.
The formula for the nth term of a Taylor series is ((f^(n)(a))/n!) * (x-a)^n.
The series is ideal for approximating a function near the point a, where a can be any real number.

Applying Series to Compound Functions:

To employ series expansions to compound functions, remember to apply the chain rule and product rule appropriately when differentiating.
For example, if you have a function f(g(x)), and you want to expand it into a Maclaurin series, you’ll need to differentiate f(g(x)) in respect to x using the chain rule, and then substitute x = 0 in to find the coefficient for that term in the series.
Notably, certain compound functions can have their series expansions simplified using known series. For instance, the series of f(e^x) could be evaluated using the series of f(x).

Error bounds in series expansions:

When function series expansions are truncated, there will be a remainder term. This inherent error can be estimated using the Lagrange Remainder theorem, which states that the error is lesser than or equal to M

((x-a)^(n+1))/(n+1)!

where M is a bound on the magnitude of the (n+1)th derivative on the interval from a to x.

Practise through calculations:

To get the most out of your revision, practice calculations manually. This includes expanding series, differentiating, substituting, and estimating the remainder.
Doing so assists in understanding the underlying foundations of series expansions, and allows you to apply them to new and unusual compound functions with confidence.
It’s advised to learn the standard expansions for commonly used functions off by heart, as these often form the building blocks of more complicated series expansions.
Finally, when working with series expansions, practising accurate algebra and arithmetic is critical. Small mistakes in these areas can lead to incorrect final answers.

These concepts form the basis of the topic and provide a foundation for further exploration. To enhance your understanding, it is encouraged to work through a variety of examples and problems, applying and extending the concepts outlined above.