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.

If x:-|R, then let [x] be the integer, x-1 < [x] <= x.

Prove that /\a:-|R /\n:-|N n*[a] <= ???

Prove that /\a:-|R /\n:-|N n*[a] <= ???

/\a:-|R /\n:-|N n*[a] <= [n*a]

page 119 in OLDTIMER

page 119 in OLDTIMER

If x:-|R, then let [x] be the integer, x-1 < [x] <= x.

Prove that /\a:-|R /\n:-|N ??? <= [n*a]/n <= a

Prove that /\a:-|R /\n:-|N ??? <= [n*a]/n <= a

/\a:-|R /\n:-|N [a] <= [n*a]/n <= a

page 119 in OLDTIMER

page 119 in OLDTIMER

If x:-|R, then let [x] be the integer, x-1 < [x] <= x.

Prove that /\n:-|N /\a(1),...,a(n):-|R +(k=1 to n) [a(k)] <= ???.

Prove that /\n:-|N /\a(1),...,a(n):-|R +(k=1 to n) [a(k)] <= ???.

/\n:-|N /\a(1),...,a(n):-|R +(k=1 to n) [a(k)] <= [ +(k=1
to n) a(k) ]

page 120 in OLDTIMER

page 120 in OLDTIMER

If x:-|R, then let [x] be the integer, x-1 < [x] <= x.

Prove that /\a,b:-|R ??? <= [a + b].

Prove that /\a,b:-|R ??? <= [a + b].

/\a,b:-|R [a] + [b] <= [a + b]

page 120 in OLDTIMER

page 120 in OLDTIMER

If x:-|R, then let [x] be the integer, x-1 < [x] <= x.

Prove that /\n:-|N /\a(1),...,a(n):-|R ??? <= [ +(k=1 to n) a(k) ].

Prove that /\n:-|N /\a(1),...,a(n):-|R ??? <= [ +(k=1 to n) a(k) ].

/\n:-|N /\a(1),...,a(n):-|R +(k=1 to n) [a(k)] <= [ +(k=1
to n) a(k) ]

page 120 in OLDTIMER

page 120 in OLDTIMER

If x:-|R, then let [x] be the integer, x-1 < [x] <= x.

Prove that /\a:-|R /\n:-|N [a] <= ??? <= a

Prove that /\a:-|R /\n:-|N [a] <= ??? <= a

/\a:-|R /\n:-|N [a] <= [n*a]/n <= a

page 119 in OLDTIMER

page 119 in OLDTIMER

If x:-|R, then let [x] be the integer, x-1 < [x] <= x.

Prove that /\a:-|R /\n:-|N 0 <= a - [n*a]/n < ???.

Prove that /\a:-|R /\n:-|N 0 <= a - [n*a]/n < ???.

/\a:-|R /\n:-|N 0 <= a - [n*a]/n < 1/n

page 121 in OLDTIMER

page 121 in OLDTIMER

Prove that every real number is the limit of a sequence of
rational numbers.

If x:-|R, then let [x] be the integer, x-1 < [x] <= x.

If a:-|R, let x[n] = [n*a]/n. Clearly, /\n:-|N x[n]:-|Q.

Recall that /\a:-|R /\n:-|N 0 <= a - [n*a]/n < 1/n,

and conclude that lim(n->oo) x[n] = a.

If a:-|R, let x[n] = [n*a]/n. Clearly, /\n:-|N x[n]:-|Q.

Recall that /\a:-|R /\n:-|N 0 <= a - [n*a]/n < 1/n,

and conclude that lim(n->oo) x[n] = a.

Let k and p be integers, k>=1, p>=0.

Let { a[n] }(n=0 to oo) be a sequence of complex numbers.

What is the radius of convergence of the power series a[n]*z^(kn+p) ?

Let { a[n] }(n=0 to oo) be a sequence of complex numbers.

What is the radius of convergence of the power series a[n]*z^(kn+p) ?

( 1/( lim_sup(n->oo) |a[n]|^(1/n) ) )^(1/k)

page 118 in OLDTIMER

page 118 in OLDTIMER

Let f : ]0,oo[ x ]0,oo[ -> |R be the function satisfying:

/\x>0,y>0 sin(arctan(y/x)) = f(x,y).

Find a formula for f(x,y) that does not involve trig functions.

/\x>0,y>0 sin(arctan(y/x)) = f(x,y).

Find a formula for f(x,y) that does not involve trig functions.

f(x,y) = y / (x^2 + y^2)

page 95 in OLDTIMER

page 95 in OLDTIMER