[sequence-real] Real number sequence := . Usually denoted as . Depending on the situation, set it to start from or from
[limit-sequence-real] Sequence limit
Operations of limits
[rational-dense-in-real] is dense in .
Proof
==>
==>
-
-
or
Proof and
==>
[geometric-series] Geometric series .
Proof ,
[geometric-series-test] Geometric series convergence test. let .
Proof
[exponential-vs-power] Exponential growth is faster than power.
Proof define
use Geometric series convergence test
[exponential-root-of-power-function]
Proof
==>
Proof
When , by
When , use
[factorial-vs-exponential-1] Factorial growth is faster than exponential.
Proof define . use Geometric series convergence test
corresponds to the number of bijections of , corresponds to the number of self-mappings of . etc. are similar
[iterated-power-vs-factorial]
Proof define . use geometric series convergence test
Comparison of growth rates, real number version
-
-
with
-
[mean-inequality] Mean inequality alias [AM-GM-inequality]
obtains <==>
Dimensionless
Proof
<==>
<==>
Use differential method to calculate the extreme value. Consider the function
First derivative
The first derivative equals zero, solving the equation yields
Second derivative
Determine the positive definiteness of the quadratic form
The multiplication factor can be extracted
All are order polynomial of , and the first-order differential being zero makes , so for judging positive definiteness, we only need to consider , the quadratic form
So at the first derivative is zero and the second derivative is (semi) positive definite, the function will not become smaller nearby, so that's the minimum, and it's
[best-multiplication-decomposition] Optimal multiplication decomposition
forall fixed
question: which makes maximum?
For each , according to the mean inequality, should be used for equal addition to obtain the maximum
When what is the equal division number , has the maximum value?
monotonically increasing
Proof
Function
-
Increasing when
-
Decreasing when
Therefore takes the maximum value near
Proof of the monotonic property of
Example . Therefore, when , 1 equal division is the best
i.e.
[natural-constant] Natural constant
Although the two limits appear to be so different in form
Proof
Binomial expansion
When is fixed, we have
For each
also
by
Therefore
converges. ==> at the tail geometric series control
[factorial-function-1]
Infinite product definition of the factorial function . Not in the subtraction direction but in the addition direction.
==>
with
Sometimes it is more convenient to use the equivalent .
To prove convergence, one method is to convert the infinite product into an infinite sum using . Using a trick.
Using Taylor expansion .
-
Using the properties of the factorial function, it can be proven that cf. Euler-reflection-formula. Here, only convergence is proven.
converges, for and for .
is called the Riemann Zeta function.
-
is the Euler gamma constant [Euler-constant]
as an additive asymptote. as a multiplicative asymptote.
Proof
let
Can use and converges.
Can also use integral estimation.
is bounded.
is monotonically decreasing.
[Euler-reflection-formula] Euler reflection formula or
Using the countable generalization of the fundamental theorem of algebra, wiki:Weierstrass_factorization_theorem
Using
The zeros of are . The zeros of are , corresponding to the zeros of
, expanding as a power series, the coefficient of is
Comparing the coefficient of in the Taylor expansion of at , which is
In particular
Thus
And we get [Wallis-formula]
[factorial-function-2]
According to Euler's insight, the integral definition of the factorial function is, for and then for (and possibly for other normed-algebra)
The two definitions of are equivalent, but this is not obvious. The extension of from to is not unique, because one can add analytic functions that take the value for to maintain the extension of , for example by adding the function
(ref-25, vol.2, sect.Euler-integral) The function sequence converges monotonically and uniformly on to . Exchange series and integral
Variable substitution can yield another integral representation
[Gaussian-integral] Variable substitution or then
We have obtained using the Euler reflection formula. It can also be obtained using the polar coordinate method
[iterated-power-vs-factorial-2]
Comparison of growth rates of factorial and tetration
so , so
Proof of
def
def
[sequence-multiplication-mean-limit] Multiplication average does not change the limit
Proof
[sequence-addition-mean-limit] Addition average
[harmonic-series-diverge] Harmonic series diverges
Proof diverges because it is not limit-distance-vanish. e.g.
[iterated-power-vs-factorial-3] [Stirling-approximation]
Tips
Taylor expansion
We know
(ref-26) The last term
So or
(ref-27) Variable substitution
The function monotonically converges to for respectively when .
Exchange series and integral, and use
For a discussion on the appearance of , also see why-pi-in-Gaussian-integral