integral de series divergents

int[x^{0}] d[x] = sum[ ( x_{n} )^{0} ]

int[x^{0}] d[x] = ...

... ( x_{0} )^{0}+...(oo)...+( x_{n} )^{0} = x

... 0^{0}+...(oo+1)...+n^{0} = oo+1 = oo

... 1+...(n+1)...+1 = n+1

int[f(x)] d[x] = sum[ f(x_{n})·( f(1)+(-1)·f(0) ) ]

int[e^{x}] d[x] = ...

... ( e^{x_{0}}(e+(-1)) )+...(oo)...+( e^{x_{n}}(e+(-1)) ) = e^{x}

... ( e^{0}(e+(-1)) )+...(oo)...+( e^{n}(e+(-1)) ) = e^{oo}+(-1) = e^{oo}

int[x] d[x] = ...

... ( x_{1} )^{1}+...(oo)...+( x_{n} )^{1} = (1/2)·x^{2}

... 1^{1}+...(oo)...+n^{1} = (1/2)·oo^{2}

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

int[x^{2}] d[x] = ...

... ( x_{1} )^{2}+...(oo)...+( x_{n} )^{2} = (1/3)·x^{3}

... 1^{2}+...(oo)...+n^{2} = (1/3)·oo^{3}

... 1^{2}+...(n)...+n^{2} = (1/6)·n(n+1)(2n+1)

int[x^{3}] d[x] = ...

... ( x_{1} )^{3}+...(oo)...+( x_{n} )^{3} = (1/4)·x^{4}

... 1^{3}+...(oo)...+n^{3} = (1/4)·oo^{4}

... 1^{3}+...(n)...+n^{3} = (1/4)·n^{2}(n+1)^{2}

conjetura:

int[x^{(1/2)}] d[x] = ...

... ( x_{1} )^{(1/2)}+...(oo)...+( x_{n} )^{(1/2)} = (2/3)·x^{(3/2)}

... 1^{(1/2)}+...(oo)...+n^{(1/2)} = (2/3)·oo^{(3/2)}

1^{(1/2)}+...(n)...+n^{(1/2)} = 2·( (1/6)·n^{(1/2)}(n^{(1/2)}+1)·(2n^{(1/2)}+1) )

int[x^{(1/3)}] d[x] = ...

... ( x_{1} )^{(1/3)}+...(oo)...+( x_{n} )^{(1/3)} = (3/4)·x^{(4/3)}

... 1^{(1/3)}+...(oo)...+n^{(1/3)} = (3/4)·oo^{(4/3)}

1^{(1/3)}+...(n)...+n^{(1/3)} = 3·( (1/4)·n^{(2/3)}(n^{(1/3)}+1)^{2} )

int[f(1/x)] d[x] = sum[ f(1/x_{n})·( f(1)+(-1)·f(0) ) ]

int[(1/x)] d[x] = ...

... (1/x_{1})+...(oo)...+(1/x_{n}) = ln(x)

... (1/1)+...(oo)...+(1/n) = ln(oo)

lim[ (1/(n+1))·( 1/ln(1+(1/n)) ) ] = lim[ ( 1/( ln( (1+(1/n))^{n} )+ln(1+(1/n)) ) ) ] = 1

conjetura:

int[(1/x)^{(1/2)}] d[x] = ...

... ( 1/x_{1} )^{(1/2)}+...(oo)...+( 1/x_{n} )^{(1/2)} = (2/1)·x^{(1/2)}

... (1/1)^{(1/2)}+...(oo)...+(1/n)^{(1/2)} = (2/1)·oo^{(1/2)}

(1/1)^{(1/2)}+...(n)...+(1/n)^{(1/2)} = ...

... (3/1)·2·(1/((n^{(1/2)}+p)(n^{(1/2)}+q)))·( (1/6)·n^{(1/2)}(n^{(1/2)}+1)·(2n^{(1/2)}+1) )

int[(1/x)^{(1/3)}] d[x] = ...

... ( 1/x_{1} )^{(1/3)}+...(oo)...+( 1/x_{n} )^{(1/3)} = (3/2)·x^{(2/3)}

... (1/1)^{(1/3)}+...(oo)...+(1/n)^{(1/3)} = (3/2)·oo^{(2/3)}

(1/1)^{(1/3)}+...(n)...+(1/n)^{(1/3)} = ...

... (4/2)·3·(1/((n^{(1/3)}+p)(n^{(1/3)}+q)))·( (1/4)·n^{(2/3)}(n^{(1/3)}+1)^{2} )

 

lim[ ( 1+...(n)...+n )^{n}·( 2/n^{2} )^{n} ] = e

lim[ ( 1^{2}+...(n)...+n^{2} )^{n}·( 6/(n^{2}·(2n+1)) )^{n} ] = e

lim[ ( 1^{3}+...(n)...+n^{3} )^{n}·( 4/(n^{3}·(n+1)) )^{n} ] = e