It seems to me that the reason
for introducing series at all in these
courses is that we want to introduce power series---Taylor
series in particular---and do calculus
with them. The importance of Taylor series lies in the
fact that these series have radii of convergence and
the functions they define inside those radii of convergence
behave very well indeed there.\\
The reason for this very nice behavior
is, of course, the fact that a power
series converges uniformly on
compact subsets of the interior
of its interval of convergence. {\em It is
this convergence---uniform on
compact subsets---that is the
mathematically important kind
of convergence\/}. Pointwise convergence,
the kind one deals with when
one deals with series of constants,
is worth very little to those
who want to do calculus with the limit
function.\\
In particular,
in a freshman calculus course
there is absolutely no {\it mathematical\/}
reason to concern ourselves with
convergence behavior at the
endpoints of the convergence interval
in a freshman calculus course.
Not only has this behavior little or
nothing to do with {\it calculus\/},
but the mathematical issues
that attend this behavior are very
difficlt. As Bill
Davis has pointed out, the
next step after considering the
convergence behavior of a real
power series at the endpoints of its
convergence interval is to consider
the convergence behavior of a
complex power series on the boundary
of its disk of convergence;
that topic is so difficult we don't
even approach it in our standard introductory
complex analysis courses at the graduate level.\\
>From the observations that the
important convergence is uniform on
compact subsets and that boundary
behavior is not only difficult but is
also irrelevant to freshman calculus,
it follows that we should examine
the former and ignore the latter.
Now current graphing technology makes
it easy for freshman to see uniform
convergence on compact sets---though they
won't realize that's what they're
seeing. Itake advantage of this
technology by examining, in some
detail, the way in which the partial
quotients for $1/(1 - x)$
approximate the fraction from which
they are obtained.\\
In my classes, we begin
the discussion by analogy
with infinite decimal expansions
obtained by division from rational
numbers, and we explore the possible
meaning of ``partial quotients'' obtained
from rational functions by formal
long division. Examination of
appropriate graphs shows that, in
every case, the partial quotients
give better and better approximations
to the rational function from which
they were obtained as the number
of terms retained in the quotient
increases---but only on certain
intervals. For mysterious
reasons, the ``approximations''
are awful in other regions. It does
appear that the approximation
not only fits the original rational
function better as the number
of terms grows, but also that the
size of the interval where the
approximation is reasonable
increases---within limits---at
the same time.\\
We look at a number of examples,
with particular emphasis on the
rational function $\varphi[x] = 1/(1 - x)$. We
also look at the relationships
between $\varphi'[x]$ and
$\int_0varphi[t]\,dt$, on the one hand,
and the polynomials obtained
by differentiation and integration
of the polynomials obtained
from $\varphi[x]$ on the other.
We calculate how many terms it
requires, for example, to guarantee
that $\sum_{k=0}will
lie within $1/10$ of $\varphi[x]$ for
all $x$ for which
$-4/5 \le x \le 4/5$. We solve similar
problems for $\sum_{k=0} x1}$
(resp., $\sum_{k=0}1}/(k + 1)$)
and $\varphi'(x)$ (resp.,
$\int_0varphi(t)\,dt$).\\
When we have acquired a wealth of
experience approximating rational
functions (and their derivatives and
integrals) by means of polynomials
obtained by truncating quotients (and
differentiating or integrating as
needed), we consider the geometric
series $\sum_{k=0}fty xtheoretically.
We establish that it converges to
$\varphi[x] = 1/(1 - x)$ in the
interval $(-1, 1)$ in such a way
that if $t \in (0, 1)$ is arbitrary
and $\epsilon > 0$ is given we can
find a natural number $N$ so that
$|\varphi[x] - \sum_{k=0}< \epsilon$
for every $x$ in $[-t, t]$
provided that $n \ge N$. We show
that, collectively, the sums behave
very badly for $|x| > 1$. Then we
establish similar facts for the
derived geometric series and the
integrated geometric series.\\
Having thoroughly examined the
geometric series, we move on to more
general series in powers of
$x$---using our analysis of the
geometric series as our paradigm.
We introduce the radius of
convergence; we discuss convergence,
differentiation, and
integration of power series and the
functions they represent all in
terms of the radius of convergence.
Generally speaking, we do not
give formal proofs here.\\
Once we have introduced the radius of
convergence, the question of the
moment is ``How can we find it?'' We
give our students not only the
Ratio and Root Tests (suitably re-stated
for direct application
to power series), but also the
Nearest-Singularity-in-the-Complex-Plane
Criterion. The latter criterion is,
of course, unprovable in
a freshman course. Depending upon
the class, we may or may not give
proofs for the Ratio or Root Tests.\\
The next step is to use the differentiability
of a function represented
by a Maclaurin series to obtain the
Maclaurin coefficients. This,
in turn, provides context in which
to practice use of our means
for finding the radius of convergence.
Ultimately, we introduce the
Taylor Polynomial with Integral Remainder,
by way of integration by
parts. We are careful to point out
that the remainder in Taylor's
Formula plays exactly the same role
that the remainder term---which we
calculated explicitly earlier by
subtracting the $n$-th partial
sum of the geometric series from
$1/(1 - x)$---plays in that
other discussion. This gives us
the tools needed to establish,
{\it e.g.\/}, that the Maclaurin
series for the sine function converges
to and represents the sine function
everywhere. (As an aside,
note that these convergence problems
also give us an excuse to spend some
time estimating integrals, and
this is a skill I have in the past
found gradates of freshman calculus sequences
to be sadly deficient in.)\\
Finally, we pursue the standard tricks
of the trade for obtaining
unknown series expansions from known
series expansions by differentiation,
multliplication, division, integration,
etc. The emphasis here is on
the notion that one can treat series as though
they were (very long) polynomials as
long as one confines one's
attentions to a region that lies
entirely inside of all of the
relevant intervals of convergence.
--Lou Talman