# 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.

### 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 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 .

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### 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,

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

is a differentiable function whose derivative is equal to the original function

Suppose , the notation is

So all the antiderivative of become a family set .

also the equation below is obviously.

### Example: Wallis product

the Wallis product for

, published in 1656 by John Wallis states that

so that:

so that:

### Simplify the Polynomial and Integral

If

and

is a proper fraction, there exists a unique representation of the fraction

in the form

and if

and

are polynomials with real coefficients and

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 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.