Chapter 5
Series and Differential Equations
The methods and results of this chapter pave the road to the students’
more serious study of “mathematical analysis,” that branch of mathe-
matics which includes calculus and differential equations. It is assumed
that the student has had a backgroud in calculus at least equivalent
with that represented either in IB mathemtics HL year 2 or AP Cal-
culus (AB). The key ideas revolving around limits will be reviewed,
leading to substantial coverage of series and differential equations.
5.1 Quick Survey of Limits
As quickly becomes obvious to even the causual learner, the study of
calculus rests in a fundamental way on the notion of limit. Thus, a
reasonable starting point in this somewhat more “advanced” study is
to be reminded of the notion of the “limit of a function asxapproaches
a(either of which might be±∞).”
5.1.1 Basic definitions
Definition. Letfbe a function defined in a neighborhood of the real
numbera(except possibly atx=a). We say that thelimitoff(x)is
Lasxapproachesa, and write
xlim→af(x) = L,
if for any real number > 0 , there is another real numberδ > 0 (which
in general depends on ) such that whenever 0 < |x−a| < δ then
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