The Abstract Of Mathematical Analysis I
1. Limits
Definition: series limit
We say that the sequence converges to
or tends to
and write
as
.
Definition: fundamental or Cauchy sequence
A sequence is called a fundamental or Cauchy sequence if for any
there exists an index
such that
whenever
and
.
Theorem: Weierstrass
In order for a nondecreasing sequence to have a limit, it is necessary and sufficient that it be bounded above.
Two important limit
Definition 3. inferior limit and superior limit
Theorem 2. Stolz
Let and
be two sequences of real numbers. Assume that
is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching
, or strictly decreasing and approaching
) and the following limit exists:
Then, the limit
Theorem 3. Toeplitz limit theorem
Supports that ,
and
if
By using
, we can quickly infer The Cauchy proposition theorem.
By using
, we can quickly infer The Stolz theorem.
Stirling’s formula
Specifying the constant in the error term gives
, yielding the more precise formula:
2. Continuity
Definition 0
A function is continuous at the point
, if for any neighbourhood
of its value
at a there is a neighbourhood
of a whose image under the mapping
is contained in
.
3. Differential calculus
Definition 0
The number
is called the derivative of the function at
.
Definition 1
A function defined on a set
is differentiable at a point x ∈ E that is a limit point of E if
, where
is a linear function in
and
as
,
.
Definition 2
The function of Definition 1, which is linear in
, is called the differential of the function
at the point
and is denoted
or
. Thus,
.
We obtain
We denote the set of all such vectors by or
. Similarly, we denote by
or
the set of all displacement vectors from the point
along the y-axis. It can then be seen from the definition of the differential that the mapping
The derivative of an inverse function
If a function is differentiable at a point x0 and its differential
is invertible at that point, then the differential of the function
inverse to
exists at the point
and is the mapping
inverse to .
The derivative of some common function formula
L’Hôpital’s rule
The theorem states that for functions and
which are differentiable on an open interval
except possibly at a point
contained in
, if

Taylor’s theorem
Let be an integer and let the function
be
times differentiable at the point
. Then there exists a function
such that ,
and,
prove:

q.e.d
remainder term
using little notation,
(The Peano remainder term)
The Lagrange form remainder term( Mean-value forms)
4. Integral
Antiderivative
Definition
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function


Suppose , the notation is
So all the antiderivative of
become a family set
.
also the equation below is obviously.
Theorem: Integration by parts
Example: Wallis product
the Wallis product for

Prove:

so that:

so that:

Simplify the Polynomial and Integral
If
and if
there exists a unique representation of the proper fraction
where and
are real numbers.
and with these formulas below:

And from that we get the recursion:
Primitives of the Form
We make the change of variable . Since:
so that
It follows that
not only
can to do this, but here are a lot of formula:
,
,
,
Integration
Riemann Sums
partition
A partition P of a closed interval ,
, is a finite system of points
of the interval such that
.
If a function is defined on the closed interval
and
is a partition with distinguished points on this closed interval, the sum
where , is the Riemann sum of the function
corresponding to the partition
with distinguished points on
.
The largest of the lengths of the intervals of the partition
, denoted
, is called the mesh of the partition.
we define:
Integral mean value theorem
If is a continuous function on the closed, bounded interval
, then there is at least one number
in
for which
The second Integral mean value theorem
If are continuous functions on the closed, bounded interval
,
is monotonous on
, then there is at least one number
in
for which
Newton-Leibniz formula
Let be a continuous real-valued function defined on a closed interval
. Let
be the function defined, s.t.
Substitution Rule For Definite Integrals
Suppose and
, s.t.
0 条评论