limit-sequence - note-math

[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

is equivalent to

Up to a difference of at most, there is

==>

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 ‘s 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

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 it’s 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 Cauchy’s 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 Euler’s 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

[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

We also need to compute the remaining constant

(ref-27) Variable substitution

Functions converge monotonically to as for respectively, because:

Interchange series and integral, and use

Discussions on the appearance of can also be seen in why-pi-in-Gaussian-integral