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.

0<x<1
Prove that lim x^n = 0, as n approaches infinity.
Put d = 1/x - 1.
Use (1+d)^n >= 1 + n*d.
page 127 in palace
Let a(n) be a sequence of nonnegative numbers.
Suppose that lim_sup a(n)^(1/n) < 1.
What can we infer about the series a(n)?
It converges.
(this is Cauchy's test)
Let a(n) be a sequence of nonnegative numbers.
Suppose that lim_inf a(n)^(1/n) > 1.
What can we infer about the series a(n)?
It diverges.
page 130 in palace
Let a(n) be a sequence of nonnegative numbers.
Suppose that lim_sup a(n)^(1/n) = 1.
What can we infer about the series a(n)?
Nothing as far as convergence goes. It can converge or diverge, the stipulated condition is inconclusive.
series 1/n diverges
series 1/n^2 converges
Let a(n) be a sequence of positive numbers.
Suppose that lim_sup a(n+1)/a(n) < 1.
What can we infer about the series a(n)?
It converges.
This is the ratio test, or D'Alembert's test.
Let a(n) be a sequence of positive numbers.
Suppose that lim_inf a(n+1)/a(n) > 1.
What can we infer about the series a(n)?
It diverges.
This is the ratio test, or D'Alembert's test.
State and prove Cauchy's Integral Test for convergence of series.
f : [1,oo) -> [0,oo) decreasing
(1) Riemann_integral_from_1_to_x_of_f converges, as x approaches infinity
(2) series f(n) converges
thesis: (1) <=> (2)
page 133 in palace
Using Cauchy's Integral Test show that the series 1/n diverges.
hint: log(x) tends to infinity, as x approaches infinity
page 133 in golden gate
Using Cauchy's Integral Test investigate the convergence of the series 1/n^A, where A is a real number.
converges for A>1
diverges for A<=1
page 135 in palace
Let a(n), b(n) be sequences of positive numbers.
(1) series b(n) converges
(2) lim_sup a(n)/b(n) = K, where K is a real number
What can we infer about the series a(n)?
converges