Is the product of an infinite infinitesimal quantity necessarily an infinitesimal quantity? Please g

First of all, I want to state that I don't know much about this issue. I know"The sum of an infinite number of infinitesimal quantities is not necessarily an infinitesimal quantity".

Let me give you an understanding of this issue.

If a<1, then the square of a is less than a.

Let a and b be infinitesimal quantities, then a < 1, and ab Since a is an infinitesimal quantity, then ab should also be an infinitesimal quantity.

Based on this, I think the conclusion can be drawn correctly.

If a<1, the infinite power of a is 0 (this is an axiom of the limit of the sequence);

Because infinitesimal quantities are less than one, the product of the infinite infinitesimal quantities must be 0.

First of all, you have to understand how to prove that a quantity is an infinitesimal quantity by comparing its absolute value to the absolute value of another infinitesimal quantity (which is derived from a theorem, the name of which is forgotten), and if it is smaller than another infinitesimal quantity, then it is an infinitesimal quantity.

So the proof on the first floor is basically correct, but you have to add the absolute value symbol, otherwise it is difficult to say if it is greater than less than if it is positive or negative, and it is perfectly proved by adding the absolute value symbol.

Proof : Let o(n) be infinitesimal then.

o(n)*o(n)*o(n)*…o(n)*1*1*……o(n)

So an infinitesimal quantity is a product and an infinitesimal quantity.

It must be. Proof: omitted.

It won't be 0

The properties of infinitesimal are:1. The sum of finite infinitesimal quantities is still infinitesimal quantities.

2. The product of a finite infinitesimal quantity is still an infinitesimal quantity.

3. The product of the bounded function and the infinitesimal quantity is the infinitesimal quantity.

4. In particular, the product of the constant and the infinitesimal is also infinitesimal.

5. The reciprocal of an infinitesimal quantity that is never zero is infinite, and the reciprocal of infinity is infinitesimal.

6. An infinitesimal quantity is not a number, it is a variable.

7. Zero can be the only constant for infinitesimal quantities.

8. The infinitesimal quantity is related to the trend of the independent variable.

Examples are as followsInfinitesimal refers to a concept in mathematical analysis, where infinitesimal quantities usually appear in the form of functions, sequences, etc., in classical calculus or mathematical analysis.

An infinitesimal quantity is a variable with the number 0 as the limit, infinitely close to 0. To be precise, when the independent variable x is infinitely close to x0 (or the absolute value of x increases infinitely), and the function value f(x) is infinitely close to 0, i.e., f(x) 0 (or f(x)=0), then f(x) is said to be an infinitesimal quantity when x x0 (or x).

The infinitesimal quantity is a function with 0 as the limit, and the speed at which the infinitesimal quantity converges to 0 can be fast or slow. Therefore, between two infinitesimal quantities, they are divided into high-order infinitesimal , low-order infinitesimal , same order infinitesimal and equivalent infinitesimal .

Hello, the analysis is as follows:

The definition function columns are as follows:

The domain is defined as: [1,+

x∈[1,2)

f1(x)=1/x, x∈[2,+∞

1,fn(x)=1, x∈[1,n)

fn(x)=x^(n-1), x∈[n,n+1)fn(x)=1/x, x∈[n+1,+∞4.Let f(x) = fn(x),x∈[1,2)

>fn(x)=1

>f(x)=∏fn(x)=1

x∈[k,k+1),k>1

fn(x)=1/x,n≤k-1

fk(x)=x^(k-1),fn(x)=1,k+1≤n

f(x)=∏fn(x)=

f1(x)*.f(k-1)(x)*fk(x)*1*1...==(1/x)*.

1/x)*x^(k-1)*1..*1...==1 so f(x) 1, so when x +, f(x) is not infinitesimal.

But for each fn(x), when x +, fn(x) is infinitesimal.

Apparently limfn(x)=0).

So the product of an infinitesimal is not necessarily infinitesimal.

Hope it helps! Give a good review, thank you!

The product of two infinitesimal is infinitesimal, so the product of an infinite infinitesimal is infinitesimal.

For example, let the function fn(x)=1 (0 x n-1).

fn(x)=x^(n-1) (n-1＜x≤n, n=1,2,3,…)fn(x)=1/x (n≤x＜+∞

then when n + for each natural number.

n has fn(x) 0, i.e. fn(x) is an infinitesimal quantity.

But their product is f(x) = 1, )fn(x) = 1, (0 x + when x +, the function f(x) is also not an infinitesimal quantity. So the product of an infinitesimal is not necessarily infinitesimal.

The proof is as follows: <>

The properties of infinitesimal are:

1. The sum of finite infinitesimal quantities is still infinitesimal quantities.

2. The product of a finite infinitesimal quantity is still an infinitesimal quantity.

3. The product of the bounded function and the infinitesimal quantity is the infinitesimal quantity.

4. In particular, the product of the constant and the infinitesimal is also the infinitesimal of the leakage.

5. The reciprocal of the infinitesimal quantity that is never zero is infinite, and the reciprocal of the infinitesimal of the chain is infinitesimal.

6. An infinitesimal quantity is not a number, it is a variable.

7. Zero can be the only constant for infinitesimal quantities.

8. The infinitesimal quantity is related to the trend of the independent variable.