# 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

, let

, s.t.

By using , we can quickly infer The

Cauchyproposition theorem.By using , we can quickly infer The

Stolztheorem.

### 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***or*

**primitive integral***of a function*

**indefinite integral**is a differentiable function whose

*is equal to the original function*

**derivative**Suppose , the notation is

So all the

of become a family set .antiderivativealso the equation below is obviously.

### Theorem: Integration by parts

### Example: Wallis product

the **Wallis product** for

, published in 1656 by

*John Wallis*states that

#### Prove:

so that:

so that:

### Simplify the Polynomial and Integral

If

and

is a

*, there exists a unique representation of the fraction*

**proper fraction**in the form

and if

and

are polynomials with

*and*

**real coefficients**

there exists a unique representation of the proper fraction

in the form

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

*with distinguished points on this closed interval, the sum*

**a partition**

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

, denoted , is called thepartitionof the partition.mesh

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.

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