Before studying the limit of a function of a real variable, it is helpful to work with a simpler and more visual case: the limit of a sequence. A sequence is an infinite, ordered list of real numbers, and the typical question is “which number do its terms approach?”. The chapter introduces the definition and the notation , recursively defined sequences, the two great families of arithmetic and geometric progressions with their summation formulae, the rigorous definition of a limit with its three quantifiers, monotone and bounded sequences with the convergence theorem, the sequence that defines Euler’s number , the link between sequences and real functions and, finally, a cross-reference to the geometric series. It is the first, discrete, step towards analysis.
Sections
- Definition and notation
- Arithmetic and geometric progressions
- Limit of a sequence
- Monotone sequences
- The sequence that defines the number e
- Relationship between sequences and real functions
- Geometric series (cross-reference)
Further reading
- The anecdote of the young Gauss
- The legend of the chessboard
- A linguistic reading of the quantifiers