continuitat-acotada y successions monótones

Teorema:

Si f(x) és continua ==> f(x) és acotada.

Demostració:

Si f(x) >] 0 ==>

(-M) < (-l)+(-s) [< l+(-s) < f(x) < l+s < M

Si f(x) [< 0 ==>

(-M) < l+(-s) < f(x) < l+s [< (-l)+s < M

Teorema:

Si ( a_{n} és monótona & a_{n} és acotada ) ==> a_{n} és convergent.

Demostració: [ Per descens ]

a_{oo} = sup(A)

a_{n} [< a_{n+1} < sup(A)

a_{n} < sup(A)

a_{oo} = inf(A)

a_{n} >] a_{n+1} > inf(A)

a_{n} > inf(A)

Teorema:

Si ( a_{n} és monótona & a_{n} és acotada ) ==> a_{n} és ben ordenada.

Demostració: [ Per inducció ]

a_{0} = min(A)

a_{n+1} >] a_{n} >] min(A)

a_{n+1} >] min(A)

a_{0} = max(A)

a_{n+1} [< a_{n} [< max(A)

a_{n+1} [< max(A)

Laboratori de problemes:

Teorema:

[1] Si ( [An][Am][ a_{n+m} >] a_{n}+a_{m} ] & a_{1} >] 0 & a_{n} és acotada ) ==> ...

... a_{n} és convergent.

[2] Si ( [An][Am][ a_{n+m} [< a_{n}+a_{m} ] & a_{1} [< 0 & a_{n} és acotada ) ==> ...

... a_{n} és convergent.

Teorema:

[1] Si ( [An][Am][ a_{n+m} >] a_{n}+a_{m} ] & a_{1} >] 0 & a_{n} és acotada ) ==> ...

... a_{n} és ben ordenada.

[2] Si ( [An][Am][ a_{n+m} [< a_{n}+a_{m} ] & a_{1} [< 0 & a_{n} és acotada ) ==> ...

... a_{n} és ben ordenada.

Teorema:

Si ( x+y = n & x+k = y^{2} & y+j = x^{2} ) ==> [Es][ x^{2}+y^{2} = s ]

Demostració:

n+k+j = s

Teorema:

[An][Ex][Ey][ x+y = n ]

Demostració:

y = (-x) || x = (-y)

u+v = n

x = u+(1/m) & y = v+( 1+(-1)·(1/m) )

x+y = n+1

Arte Matemático:

Método de Exposición de función:

Sea R una relación ==>...

... Si x R y ==> x R f(y)

Arte:

[1] [En][ x+y = n <==> x+y = 1 ]

[2] [En][ x+y = (1/n) <==> x+y = 1 ]

Exposición:

x+y = (n/n) = 1

n = 1

Arte:

[1] [Em][ (-m) < a_{n} < m <==> (-1) < a_{n} < 1 ]

[2] [Em][ (-1)·(1/m) < a_{n} < (1/m) <==> (-1) < a_{n} < 1 ]

Exposición:

(-1) = (-1)·(m/m) < a_{n} < (m/m) = 1

m = 1

Arte:

[Ex][ f(x) = x^{n} <==> f(x) = nx^{n+(-1)} ]

Exposición:

f(x) = d_{x}[x^{n}] = nx^{n+(-1)}

f(x) = int[ nx^{n+(-1)} ]d[x] = x^{n}

x = n

Arte:

[En(k)][ (1/ln(n)) = (1/k) <==> (1/e^{k}) = (1/n) ]

Exposición:

(ln(n)/e^{k})·(1/ln(n)) = (1/k)·(k/n)

(e^{k}/ln(n))·(/e^{k}) = (1/n)·(n/k)

n(k) = e^{k}

Arte:

[En][ e^{n} = (n+1)+(-n) <==> 2 = ((1/n)+1)+(-1)·(1/n) ]

Exposición:

(2/e^{n})·e^{n} = (1/n^{2})·( (n+1)+(-n) ) = ((1/n)+1)+(-1)·(1/n) & ...

... f(1/n^{2}) = 1+(-1)·(1/n^{2})

(e^{n}/2)·2 = n^{2}·( ((1/n)+1)+(-1)·(1/n) ) = (n+1)+(-n) & ...

... f(n^{2}) = 1+(-1)·n^{2}

n = 0

Arte:

[En(k)][ ln(n) = ln( ln(n) )+O( n·ln(n) ) ]

Exposición:

n·ln(n) = ln(n)+O( n·ln(n) )

0^{4} [< 0 [< 1+(-1)·(1/n) [< 1

n(k) = e^{k}

k = ln(k)+O( e^{k}·k )

0^{4} [< (1/e^{k})·( 1+(-1)·(ln(k)/k) ) [< (1/e) [< 1

Arte:

[En(k)][ sum[ k [< n ][ (ln(k)/k) ] = ln(n)+O(n) ]

Exposición:

n = sum[ k [< n ][ (k/k) ] = ln(n)+O(n)

0^{4} [< 0 [< 1+(-1)·(ln(n)/n) [< 1 [< e

n(k) = e^{sum[ k [< n][ (ln(k)/k) ]+(-1)}

0^{4} [< (e/e^{oo}) [< (e/e^{sum[ k [< n][ (ln(k)/k) ]}) [< e

Arte:

[En(k)][ sum[ k [< n ][ (1/k) ] = ln(n)+O(n) ]

Exposición:

n = sum[ k [< n ][ (1/1) ] = ln(n)+O(n)

0^{4} [< 0 [< 1+(-1)·(ln(n)/n) [< 1

n(k) = e^{sum[ k [< n][ (1/k) ]+(-1)}

0^{4} [< (e/e^{oo}) [< (e/e^{sum[ k [< n][ (1/k) ]}) [< 1

Arte:

[En(k)][ prod[ k [< n ][ (1+(-1)·(1/k)) ] = ( 1/ln(n) )·( 1+O( ln(n) ) ) ]

Exposición:

1 = prod[ k [< n ][ (2+(-1)) ]  = prod[ k [< n ][ (1+(-1)) ] = ( 1/ln(n) )·( 1+O( ln(n) ) )

1 = (1/ln(n))·( 1+O( ln(n) ) )

1+(-1)·(1/ln(n)) [< 1

0 [< 1+(-1)·(1/ln(n+1)) [< 1

n(k) = e^{( prod[ k [< n ][ (1+(-1)·(1/k)) ] )^{(-1)}·k}

0 [< ( prod[ k [< n ][ (1+(-1)·(1/k)) ] )·( 1+(-1)·(1/k) ) [< 1

Arte:

[Ex][ (x^{2}/6) = sum[ (1/k^{2}) ] <==> (x^{3}/5·2·2) = sum[ (1/k^{3}) ] ]

Exposición:

g(x^{2}/6) = (6x/20)·(x^{2}/6) & f(a_{k}) = a_{k}·(1/k)

g(x^{3}/20) = (20/6x)·(x^{3}/20) & f(b_{k}) = b_{k}·k

x = pi

Arte:

[Ex][ (x^{4}/90) = sum[ (1/k^{4}) ] <==> (x^{5}/11·3·3·3) = sum[ (1/k^{5}) ] ]

Exposición:

g(x^{4}/90) = (90x/297)·(x^{4}/90) & f(a_{k}) = a_{k}·(1/k)

g(x^{5}/297) = (297/90x)·(x^{5}/297) & f(b_{k}) = b_{k}·k

x = pi

Arte:

[Ex][ (x^{6}/945) = sum[ (1/k^{6}) ] <==> (x^{7}/17·7·5·5) = sum[ (1/k^{7}) ] ]

Exposición:

x = pi

Arte:

[Ex][ (x^{8}/9450) = sum[ (1/k^{8}) ] <==> (x^{9}/11·3·3·3·5·5·2·2) = sum[ (1/k^{9}) ] ]

Exposición:

x = pi