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.

Let a(n), b(n) be sequences of positive numbers.
(1) series b(n) diverges
(2) lim_inf a(n)/b(n) is greater than zero
What can we infer about the series a(n)?
diverges
Let a(n), b(n) be sequences of positive numbers.
(1) series b(n) diverges
(2) lim_inf a(n)/b(n) = K, where K is a real number
What can we infer about the series a(n)?
If K>0, then the series a(n) diverges.
If K=0, then we cannot infer anything.
a(n)=1/n^2, b(n)=1/n, a(n) converges
a(n)=1/n, b(n)=1/log(n), a(n) diverges
and in both cases the limit is zero
Let a(n) be a sequence of nonnegative numbers.
Suppose lim_sup a(n)^(1/n) > 1.
What can we infer about the series a(n)?
diverges
(This is Cauchy's test.)
(1) the series a(n) is bounded, it is a series of complex numbers
(2) b(n) tends to zero
(3) b(n) is monotonic
What can we infer about the series a(n)b(n)?
converges
This is Dirichlet-Abel test.
page 137 in palace
Let a(n) be a sequence of complex numbers, b(n) of real numbers.
(1) the series a(n) converges
(2) b(n) is monotonic
(3) b(n) is bounded
What can we infer about the series a(n)b(n)?
converges
Use Dirichlet-Abel test.
Hint:
(1) the series a(n) is bounded
(2) {b(n)-g} is a monotonic sequence which tends to zero
[ g(f(x)) ]' = ?
Derive the formula.
[ g(f(x)) ]' = g'(f(x)) * f'(x)
Warning: Do not divide by zero!
Consider a convex function defined on an open interval (a,b). Does this function have to have limits at points a and b?
Yes.
Hint: a monotonic function has a limit.
The limits may be improper, though.
Let s be a fixed real number.
Knowing the derivative of the exponential function,
derive the formula (x^s)' = ?
Use log.
page 190 in 1st analysis
Consider a series of positive numbers.
Suppose that lim_sup a(n+1)/a(n) > 1.
Does the series have to diverge?
NO.
1/2 + 1/3 + (1/2)^2 + (1/3)^2 + (1/2)^3 + (1/3)^3 + ...
converges
Test the series (n!)/(n^n) for convergence.
converges, the ratio test