The proof of the limit of higher number lim n^(1/n)=1 How to prove it?

lnn=nln(n^(1/n))=nln(1+n^(1/n)-1)=n{(n^(1/n)-1)-1/2(n^(1/n)-1)^2+...}
The main part of lnn is n (n ^ (1 / N) - 1), so the above limit holds

The proof limit of higher numbers
Prove Lim 0.99... 9 = 1, note: n 9!
n→∞

Xn＝0.99...9＝9/10＋9/10^2＋…… ＋9/10^n＝1－1/10^n
For any small positive number ε (ε < 1), if | XN-1 | = 1 / 10 ^ n < ε, as long as n > LG (1 / ε), there exists a positive integer n > LG (1 / ε). When n > N, | XN-1 | < ε
So Lim 0. 99... 9 = 1

At the beginning of learning integral, can I ask if discontinuous functions can also be definite integral?

In fact, the integral is to find the area of the graph. If the function is not continuous, it can be integrated piecewise. In an integral symbol, the function can only be continuous

What is the meaning of triple integral integrand function f (x, y, z)

When f (x, y, z) denotes density, triple integral denotes the mass of space object

An example of finding f (x) satisfies the following conditions: the function f (x) is defined on [a, b] and | f (x) | is integrable on [a, b], but f (x) is not integrable on [a, b]
Find the f (x) function in a high number,

Do you need f not r-integrable or l-integrable
|F (x) | is integrable on [0,1], but f (x) is not integrable on [0,1]
The functions in higher numbers are simpler
In [0,1], we define: F (x) = 1, X is a rational number; f (x) = - 1, X is an irrational number
Then we can get that | f (x) | is integrable on [0,1], but f (x) is not integrable on [0,1]
Note: if f is integrable, then the discontinuities of F must be finite
ps：
|F (x) | is integrable on [0,1], but f (x) is not integrable on [0,1] (it is also applicable to the case that R is not integrable)
In [0,1], we put the numbers satisfying X-Y is rational numbers in the same box as an equivalent class
Then take one element from each box to form a new set a
Define f (x) = 1, X belongs to a; f (x) = - 1, X does not belong to a
Then we can get that | f (x) | is integrable on [0,1], but f (x) is not integrable on [0,1]
In this conclusion, you need to prove that you only need to know that the set a is unmeasurable, and then use the definition of integrability to get f is not integrable

If f (x) defined on R is not a constant function, f (x-1) = f (x + 1), f (1 + x) = f (1-x), then f (x)
Odd function even function odd function and even function non odd non even function

F (- x-1) = f (- x + 1) = f (1-x) = f (1 + x) f (- x-1) = f [- (x + 1)] = f (1 + x), so f (x) is an even function

F (x) is a function with period T. is the definite integral of F (x) from 0 to a equal to the definite integral of F (x) from t to a + T? Why

Equal to
Because f (x) is a function with period T, so
f(x-T)=f(x)
So the definite integral of F (x) from t to a + T is equal to the definite integral of F (x-t) from t to a + t,
Let t = x-t, then the integral limit becomes from 0 to a, DX = DT,
The definite integral of F (x-t) from t to a + T is equal to the definite integral of F (T) from 0 to a
In conclusion, the definite integral of F (x) from 0 to a is equal to the definite integral of F (x) from t to a + t

Let f (x) be a continuous function with period T, then the value of definite integral ∫ (a, a + TF (x)) DX A: is independent of T, B: is independent of a and T, C: is independent of A

Let's start with C. for example, | SiN x | is a periodic continuous function. If we take pie as a period and two pie as a period, the result is different, but it has nothing to do with the value of A

A property of definite integral of periodic function is not clear
The necessary and sufficient condition for the f (T) DT of ∫ upper limit x lower limit 0 to take t as the period is that ∫ upper limit t lower limit 0 f (T) DT = 0 (that's it. I don't understand why the definite integral 0 is the algebraic sum of areas. It's hard to find that the definite integral is equal to 0, even if it's any periodic function sequence such as y = SiNx + 5, how can the definite integral be 0?)

Of course, this is for trigonometric functions, regardless of intercept
If it is SiNx + 5, the integral will be taken separately, and SiNx can simplify the integral according to periodicity
In a period, the net area sum of trigonometric function curve and X axis is always 0

Is the product of infinitesimal and bounded function an infinitesimal right or wrong

This conclusion is correct

The proof of the limit of higher number lim n^(1/n)=1 How to prove it?