Math note (3)-sums (2)

This part focuses on infinite, finite points, and infinite and solving. The author introduces factorial-like symbols, cleverly applies and finite points, and uses them to greatly simplify some complex summation. For infinity and, it is described in a different way than in advanced mathematics.

2.6 finite and infinite Calculus

1. Introduction of limited and unlimited points and symbols

Infinite points are based on the derivation symbol D:

Finite points are based on Difference symbols:
(2.42)

We introduce two new symbols:

• X to the m falling:
  (2.43)
  (2.43)
• X to the m rising:
  (2.44)

So it is easy to know its satisfying nature:

• (2.45)
•  
  (2.46)

C is an arbitrary function where p (x) satisfies p (x + 1) = p (x ).

•  
  (2.47)

So what exactly does the above formula mean ·? We can obtain the following through some special upper and lower limits:

1. So that B =:
2. So that B = a + 1:
3. Add 1 to B:

Through the above analysis and observation and mathematical induction, We can get:

•  
  (2.48)

Based on the analysis of B <A, we can obtain any integer A, B, C:

•  
  (2.49)

From 2.45, 2.48, 2.48, we can easily obtain that for m, n is greater than or equal to 0:

•  
  (2.50)

2. sum according to 2.50 Application

1. You can obtain the following information:
2. You can obtain the following information:
3. You can obtain the following information:

3. Expansion of negative numbers

Define negative falling powers (M> 0 ):
(2.51)

We can also obtain the property of a similar index:
(2.52)

So it is also true when M is a negative value of 2.50. So what should M be when M =-1 ?, It is not difficult to get it. In summary, we can get:

•  
  (2.53)

4. Step-by-Step summation

Inspired by step-by-step points, we can explore the distribution summation:

 
(2.54)

If we make ef (x) = f (x + 1), we can get the product difference formula:

(2.55)

Partial and formula (the sum of infinity and sum accumulated on both sides ):

 
(2.56)

We apply it to sum:

Finally, an example of harmonic series is obtained,
So. So:

Finally, we can get:

 
(2.57)

2.7 infinite sums

1. Non-Negative Definition:

The infinite Sum definition must be careful, otherwise it may fall into a self-contradictory situation: if we make it, then, in the end, we will get t =
-1. This is obviously a problem. It is not difficult to find a reasonable definition: for constant A: For all finite sets (K can enable an infinite set), we define that all the conditions that satisfy the minimum of the preceding inequality are, if such a does not exist, let's say. According to this definition, for non-negative items, its infinite sum can be expressed as follows:

2. According to the above definition, we can easily calculate some infinity and such:

Consider one question:
(2.58) If we change this way: Then we can get the result 1. If we move the parentheses one by one: we also get 1 because the result in the parentheses is :. When we shift the right: this is also the case (it will be proved later), which leads to inconsistency.

3. Definitions of positive and negative situations

Any real number can be expressed as follows:, then we define infinite
Sum:
(2.59) if the two items on the Right of formula 2.59 are infinite, we say it is meaningless. And, when there is a time limit for the sum,
We mean absolute convergence (absolutely
Convergent), when infinity and limited time,
Let's say that divergence is positive infinity, and vice versa, divergence is negative infinity. If it is infinitely large and infinitely infinite, there is no solution. In the same way, we can extend it to real numbers. For the sum of absolute convergence, it satisfies the distributive, associative, and commutative mentioned above.
Laws.

4. Absolute Convergence and computing Sequence

Formally speaking, if the combination of J and both subscripts causes absolute convergence to A, there will be real numbers and