# 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.

lim x*x * sin(y) / (x*x + y*y)
as (x,y) -> (0,0)
= 0
hint: |sin(y)|<=|y|; |xy/(x^2+y^2)| is bounded
lim x*x + y*y / (x*x*x*x + y*y*y*y)
as (x,y) -> (0,0)
= oo
hint: lim xy/(x+y) = 0 as (x,y) -> (0,0)
better hint: x^4 + y^4 = (x^2 + y^2)^2 - 2*x*x*y*y
lim x*x / (x*x + y*y)
as (x,y) -> (oo,oo)
doesn't exist
g : |R x |R -> |R
g(x,y) = sin(xy) / x for (x != 0)
g(0,y) = y
Is g continuous?
yes
z[n] = x[n] + i*y[n], ( x[n], y[n] are real sequences )
(1) series |z[n]| converges
(2) series |x[n]| and series |y[n]| both converge
Show that (1) <=> (2)
_______
Prove that
series 1 / ( (n+z) * log(n) * log(n) )
converges absolutely for all z:-C.
hint1: |z+n|/n -> 1
hint2: series 1/(n*log(n)*log(n)) converges
Show that the series z^n / (n*n) converges uniformly on |z|<=1.
______
Show that the series 2^n / ( z^n + z^(-n) )
converges uniformly on 0 < |z| <= r < 1/2.
hint: |a+b| >= | |a| - |b| |
|z| = 1 and (z != 1)
Show that the series (z^n)/n converges.
hint: series z^n is bounded
page 15 in OLDTIMER
Give an example of a power series that converges
for all points |z| = R except one point.
series (z^n)/n