[sequence_real] Real sequence := . Usually denoted as . Depending on the situation, start from or
[limit_sequence_real] Limit of sequence
Operations on limits
[rational_dense_in_real] Density of 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 and
[factorial_vs_exponential_1] Factorial grows faster than exponential.
Proof define . Use geometric series convergence test
corresponds to the number of bijections from to itself, corresponds to the number of self-maps from to itself. and similar
[iterated_power_vs_factorial]
Proof define . Use geometric series convergence test
Comparison of growth rates, real number version
-
Power vs Exponential:
-
Exponential vs Factorial with
-
Factorial vs Tetration
[mean_inequality] Mean inequality alias [AM_GM_inequality]
Get <==>
Dimensionless
Proof
<==>
<==>
Calculate the maximum value using differential methods. Consider proving
Since the problem is homogeneous, we only need to consider the case where , and prove that
Let
If some , then . So we only need to consider
The boundary of the intersection of and is when some , and at this point . We only need to consider points having differential zero in interior without boundary.
Using the Lagrangian multiplier method. Let the differential of be zero on the tangent space of the surface , which is equivalent to the gradients and being collinear.
First-order differential
Collinearity implies
According to , we get , which gives . At this point .
Second derivative of
Determine the positive definiteness of the quadratic form
multiplication factor can be factored out
are all degree term polynomials of , and the first-order derivative being zero makes so for judging positive definiteness we only need to consider , the quadratic form
so at the first-order derivative is zero and the second-order derivative is (semi-)positive definite, the function will not decrease nearby, so there is a local minimum, and it is
[best_multiplication_decomposition] optimal multiplication decomposition
forall fixed
question: which makes maximal?
For each , according to the AM-GM inequality, the maximum of should be achieved by using equal parts for addition.
What value of equal division number maximizes ?
The function is monotonically increasing
Proof
The function
-
Increasing when
-
Decreases when
So takes its maximum near
Proof of โs monotonicity
Example . So when , decreases, partition is optimal
i.e.
[natural_constant] Natural constant
Even though the forms of the two limits look so different
Proof
Binomial expansion
When is fixed, we have
For each
also
by
So
converges. ==> In the tail geometric series control
[factorial_function_1]
Infinite product definition of the factorial function . Not by subtraction direction but by addition direction
==>
with
Sometimes itโs more convenient to use the equivalent
To prove convergence, one method is to convert the infinite product into an infinite sum using . Use a trick.
Use Taylor expansion
-
Using the properties of the factorial function, we can prove cf. Euler_reflection_formula. Here we only prove convergence
converges, for and for
is called the Riemann Zeta function
-
is the Euler gamma constant [Euler_constant]
as additive asymptotic. as multiplicative asymptotic
Proof
let
You can use and converges
You can also use integral estimation
is bounded
is monotonically decreasing
[Euler_reflection_formula] Euler reflection formula or
converges absolutely := converges, equivalently converges, since , , equivalently converges.
absolute convergence implies convergence
So converges
Use
The zeros of are . The zeros of are , corresponding to the zeros of
(asymptotically) is controlled by . Finite product, after expanding the multiplication, is a polynomial . On it is controlled by , giving compact uniform convergence, it can be proven that the -th derivative of the sequence converges to the -th derivative of , thus it can be proven that the coefficients of the power function of the sequence converge to the coefficients of the power function of
-
Using Cauchyโs integral formula
- Or, in the case of purely real numbers, prove that is controlled, and then
- Infinite product theory in complex analysis
-
Or, from using Fourier transform integral (ref-25, vol.2)
for
let
The series is controlled and convergent. Integral, exchange integral and series
, expanding into a power series, the coefficient of is
Compare with the coefficient of in the Taylor expansion of at , which is
Specifically
Thus
And we get [Wallis_formula]
[factorial_function_2]
According to Eulerโs insight, the integral definition of the factorial function is, for and then for (and possibly for other normed-division-algebra)
The two definitions of are equivalent, but this is not obvious. The extension of from to is not unique, because one can add an analytic function that takes the value at to maintain the extension to , for example, adding the function
(ref-25, vol.2, sect.Euler-integral) The function sequence is monotonically increasing convergent and uniformly convergent on to . Interchange the series and the integral
Variable substitution can yield another integral representation
[Gaussian_integral] Variable substitution or then
We already obtained using the Euler reflection formula. We can also use the polar coordinate method
[iterated_power_vs_factorial_2]
Comparison of the growth rates of factorial and hyper-exponentiation
so , so
Proof of
def
def
[sequence_multiplication_mean_limit] Multiplication mean does not change the limit
Proof
[sequence_addition_mean_limit] Addition mean
[harmonic_series_diverge] Harmonic series diverges
Proof diverges by it is not limit-distance-vanish. e.g.
[iterated_power_vs_factorial_3] [Stirling_approximation]
Usage Tips
Taylor expansion so
We know . Thus
(ref-26) The last term is absolutely convergent
So can be decomposed into a term converge to constant
plus another term
so or
or
We also need to compute the remaining constant
(ref-27) Variable substitution
so
Functions converge monotonically to as for respectively, because:
Interchange series and integral, and use
So
So
Discussions on the appearance of can also be seen in why_pi_in_Gaussian_integral