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 .
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
By using , we can quickly infer The Cauchy proposition theorem.
By using , we can quickly infer The Stolz theorem.
Specifying the constant in the error term gives , yielding the more precise formula:
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
is called the derivative of the function at .
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 , .
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 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
The theorem states that for functions and which are differentiable on an open interval except possibly at a point contained in , if
Let be an integer and let the function be times differentiable at the point . Then there exists a function such that ,
using little notation, (The Peano remainder term)
The Lagrange form remainder term( Mean-value forms)
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.
Theorem: Integration by parts
Example: Wallis product
the Wallis product for
, published in 1656 by John Wallis states that
Simplify the Polynomial and Integral
is a proper fraction, there exists a unique representation of the fraction
in the form
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:
It follows that
not only can to do this, but here are a lot of formula:
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.
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
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.