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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 |