Theory of Statistics - George Mason University

0.0 Some Basic Mathematical Concepts 649
Let s = ∞
i=1 xi, and for any ɛ > 0, let N be such that for n > N, |Sn −s| < ɛ.
We can now write the left-hand side of the equation above as
Sn − 1
N−1
(ai+1 − ai)Sk − 1
n−1
(ai+1 − ai)Sk
an
i=1
an
i=1
N−1
an
i=N
= Sn − 1
N−1
(ai+1 − ai)Sk − 1
n−1
(ai+1 − ai)s − 1
n−1
(ai+1 − ai)(Sk − s)
= Sn − 1
an
i=1
an
i=N
(ai+1 − ai)Sk − an − aN
an
s − 1
an
i=N
an
i=N
n−1
(ai+1 − ai)(Sk − s) .
Now, consider limn→∞. The first term goes to s, which cancels with the third
term. The second term goes to zero (because the sum is a fixed value). Since
the sequence {ai} is nondecreasing, the last term is bounded by an−aN
ɛ, which
an
is less than or equal to ɛ, which was any positive number.
In Section 0.0.9.3 beginning on page 671, we list some additional ways of
determining whether or not a series converges.
0.0.5.6 Real Functions
Real-valued functions over real domains are some of the most important mathematical
objects. Here we will discuss some of their simpler characteristics.
In Section 0.1.5 we will consider some properties in more detail and in Sections
0.1.6 through 0.1.13 we will consider important operations on real functions.
We will often consider the domain of a function to be an interval [a, b], or
if the domain is in IR k , to be a rectangle [a1, b1] × · · · × [ak, bk], and many
concepts relating to a finite partitioning P of that domain. The partition may
be defined by the sets {Ii : i ∈ P }, or especially in the case of [a, b] in IR, by
(a = x0, x1, . . ., xn = b).
Important properties of functions include continuity, differentiability, integrability,
and shape. The first three of these properties, which are defined
in terms of limits, are essentially dichotomous, but they have various levels
depending on whether they hold over certain subdomains of the function.
Taylor’s Theorem
One of the most important and useful facts in analysis is Taylor’s theorem.
We state the theorem here for scalar-valued real functions of a scalar real
variable, but similar results hold for more general functions.
Theorem 0.0.11 (Taylor’s theorem)
Let f be a function defined on D ⊆ IR, let n be a positive integer, suppose that
the (n − 1) th derivative of f is continuous on the interval [x0, x] ⊆ D, and
suppose that the n th derivative of f exists on the interval ]x0, x[. Then,
Theory of Statistics c○2000–2013 James E. Gentle