Math ASCII Notation Demo

Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Test the series 1/n * sin(pi/4 * n) for convergence.
converges, Dirichlet-Abel test
Test the series (-1)^n / (n - (-1)^n) for convergence.
The series converges.
Define S[n] = +(k=1 to n) (-1)^k / (k - (-1)^k).
Prove that lim(n->oo) S[2n] = log(2).
Prove that lim(n->oo) S[2n+1] = log(2), too.
See page 83 in OLDTIMER about this log(2).
The number log(2) is not essential here.
The series a(n) converges.
Does the series a(n)*a(n) have to converge?
NO.
a(n) = (-1)^n / sqrt(n)
Let a(n) be a sequence of positive numbers.
Suppose that the series a(n) converges.
Does the series a(n)*a(n) have to converge?
Yes.
Let a(n) be a sequence of positive numbers.
Suppose that the series a(n) converges.
Does the series n*a(n) / (n+a(n)) have to converge?
Yes.
n/(n+a(n)) is less than 1
Let f be a convex function defined on an interval on the real line. Does it have to be continuous?
NO.
f:[0,1] -> R
f(1) = 1
f(x) = 0
This function is convex but is not continuous.
What is a contraction?
(X,d) metric space
f : X -> X
f is a contraction iff there exists 0 < K < 1 such that for every x,y belonging to X we have d( f(x), f(y) ) <= K * d(x,y).
State and prove Banach's Contraction Principle.
In a complete metric space every contraction has a unique fixed point.
(x is a fixed point <=> f(x) = x)
Also known as Banach's Fixed Point Theorem.
page 56 in golden gate
Prove that in a complete metric space, a countable intersection of a collection of open dense sets is dense.
page 133 in golden gate
Baire Category Theorem
Prove that if a complete metric space is written as a union of countably many closed sets, then at least one of those closed sets contains a ball.
Use the fact that in a complete metric space, a countable intersection of a collection of open dense sets is dense.
page 133 in golden gate