Preface

This course is intended for 2nd year students of Material Sciences (MS). It has nine (09)
main chapters, which expose the methods for calculating in…nite sums such as numer-
ical sequences, sequences of functions, series of functions, integer series, and di¤ equations,...etc.
The aim of this course is to generalize the notion of …nite sum of terms by studying how the
latter behaves when we consider an in…nite succession of terms. The key will be to consider these
in…nite sums, also called series, as the limit of sequences. In other words, when we remember
the course on sequences, it will be easier to assimilate the course on the series This is why the
…rst two chapters concerning reminders should not be neglected.
One of the key points of this course will be the study of Fourier series whose applications are quite
numerous in other areas of mathematics (notably di¤erential equations and partial di¤erential
equations). To reach the chapter concerning Fourier series, however, it will be necessary to
take a short path which will take us there in a less abrupt way. As we wrote above, we will
recall the structure of R, then the notion of sequences in R or C. We will then consider the
series in their generality, then the sequences and series of functions, to then move on to integer
series, to functions developable in integer series and …nally Fourier series. We can then solve
some di¤erential equations using this theory. The objective of the other chapters of the course
will be to solve di¤erential equations using Laplace transforms. This mathematical tool cannot
be applied rigorously without a little preliminary work on integrals depending on a parameter.