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)

as (x,y) -> (0,0)

= 0

hint: |sin(y)|<=|y|; |xy/(x^2+y^2)| is bounded

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)

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

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)

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?

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)

(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.

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

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.

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.

Show that the series (z^n)/n converges.

hint: series z^n is bounded

page 15 in OLDTIMER

page 15 in OLDTIMER

Give an example of a power series that converges

for all points |z| = R except one point.

for all points |z| = R except one point.

series (z^n)/n