For some clear notes and examples on Maclaurin Series see Differentiation Applications 5 from AJ Hobson’s Just the Maths (individual pdfs hosted by UEA) (or a complete pdf from the Math Centre: AJ Hobson’s ‘Just the Maths’).
For further notes, designed for students, try the HELM Project. This project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include very clear worked examples. The complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.
Use Desmos to see how Maclaurin series approximate functions – use the sliders to use an increasing number of terms of the series. Have a look at f(x) = cos x.
Try these further Desmos pages:
f(x) = sin x f(x) = sinh x f(x) = cosh x