grup producte integral

definició
f_{1}(x) [o(x)o] ...(n)... [o(x)o] f_{n}(x) = ∫ [ d_{x}[f_{1}(x)]·...·d_{x}[f_{n}(x)] ] d[x]
d_{x}[ f_{1}(x) [o(x)o] ...(n)... [o(x)o] f_{n}(x) ] = d_{x}[f_{1}(x)]·...·d_{x}[f_{n}(x)]


∫ [ (1/d_{x}[f(x)]) ] d[x] = ( f(x) )^{[o(x)o](-1)}


( f(x) [o(x)o] g(x) ) [o(x)o] h(x) = f(x) [o(x)o] ( g(x) [o(x)o] h(x) )
∫ [ ( d_{x}[f(x)]·d_{x}[g(x)] )·d_{x}[h(x)] ] d[x] = ∫ [ d_{x}[f(x)]·( d_{x}[g(x)]·d_{x}[h(x)] ) ] d[x]
f(x) [o(x)o] ( f(x) )^{[o(x)o](-1)} = x
∫ [ d_{x}[f(x)]·( 1/d_{x}[f(x)] ) ] d[x]=x
f(x) [o(x)o] x = f(x)


( f(x)  )^{[o(x)o]n} = ∫ [ d_{x}[f(x)]^{n} ] d[x]


( e^{x} )^{[o(x)o](-1)} = (-1)e^{-x}
( x^{n} )^{[o(x)o](-1)} = (1/n)·(1/((-n)+2))·x^{(-n)+2}
( bx )^{[o(x)o](-1)} = (1/b)·x
( ax^{2}+bx )^{[o(x)o](-1)} = (1/2a)·ln(2ax+b)
( ax^{3}+bx^{2}+cx )^{[o(x)o](-1)} = ( ln(3ax^{2}+2bx+c) [o(x)o] (1/6a)·ln(6ax+2b) )
( ax^{4}+bx^{3}+cx^{2}+dx )^{[o(x)o](-1)} =...
... ( ln(4ax^{3}+3bx^{2}+2cx+d) [o(x)o] ln(12ax^{2}+6bx+2c) [o(x)o] (1/24a)·ln(24ax+6b) )

funcions eliptiques

k_{n!}=k_{1}·k_{2}·...·k_{n}


cos[n](t) = ∑ ( ( (-1)^{k_{n!}}/(2(k_{n!}))! )·t^{(2(k_{n!}))})


sin[n](t) = ∑ ( ( (-1)^{k_{n!}}/(2(k_{n!})+1)! )·t^{(2(k_{n!})+1})


x^{n+1}+y^{n+1} = n^{n+1} <==> ( x=cos[n](t) <==> y=sin[n](t) )
x^{n+1}+y^{n+1} = 1 <==> ( x=(cos[n](t)/n) <==> y=(sin[n](t)/n) )


cos[n](0)=n
sin[n](0)=0
sin[n](pi/2) = n
cos[n](pi/2) = 0
sin[n](pi) = n·( 1+(-1)^{n} )^{(1/(n+1))}
cos[n](pi) = (-n)
sin[n]( (-1)(pi/2) ) = (-n)
cos[n]( (-1)(pi/2) ) = n·( 1+(-1)^{n} )^{(1/(n+1))}


sin[n](pi/4) = ( n/2^{(1/(n+1))} )
cos[n](pi/4) = ( n/2^{(1/(n+1))} )
sin[n]( (-1)(pi/4) ) = ( (-n)/2^{(1/(n+1))} )
cos[n]( (-1)(pi/4) ) = ( (n/2^{(1/(n+1))})·(2+(-1)^{n})^{(1/(n+1))} )
sin[n](3pi/4) = ( (n/2^{(1/(n+1))})·(2+(-1)^{n})^{(1/(n+1))} )
cos[n](3pi/4) = ( (-n)/2^{(1/(n+1))} )
sin[n]( (-1)(3pi/4)) = ( (-1)^{n}(n/2^{(1/(n+1))}) )
cos[n]( (-1)(3pi/4) ) = ( (-1)^{n}(n/2^{(1/(n+1))}) )


d_{x}[f(x)]=( 1+(-1)( f(x) )^{n+1} )^{(1/(n+1))} <==> f(x)=(sin[n](x)/n)
d_{x}[g(x)]=(-1)·( 1+(-1)( g(x) )^{n+1} )^{(1/(n+1))} <==> g(x)=(cos[n](x)/n)


d_{x}[f(x)]=a·( 1+(-1)( f(x) )^{n+1} )^{(1/(n+1))} <==> f(x)=(sin[n](ax)/n)
d_{x}[g(x)]=(-a)·( 1+(-1)( g(x) )^{n+1} )^{(1/(n+1))} <==> g(x)=(cos[n](ax)/n)





x^{(n+1)·p}+y^{(n+1)·q} = n^{n+1} <==> ... 
... ( x = ( cos[n](x) )^{(1/p)} & y = ( sin[n](x) )^{(1/q)} )

producte integral y ecuacions integrals

f(x) [o(x)o] g(x) = int[ d_{x}[f(x)]d_{x}[g(x)] ]d[x]


int[g(x)]d[x] [o(x)o] int[f(x)]d[x] = int[h(x)]d[x] <==> f(x)=( h(x)/g(x) )
int[g(x)]d[x] [o(x)o] ( int[f(x)]d[x]+f(x) ) = int[h(x)]d[x] <==> f(x)=e^{(-x)}·int[( h(x)/g(x) )·e^{x}]d[x]
int[g(x)]d[x] [o(x)o] ( int[f(x)]d[x]+d_{x}[f(x)] ) = int[h(x)]d[x] <==> ...
...f(x)=int[ cos(s)·int[( h(x)/g(x) )·cos(x)]d[x] ]d[x]+int[ sin(s)·int[( h(x)/g(x) )·sin(x)]d[x] ]d[x]


int[g(x)]d[x] [o(x)o] int[f(x)]d[x] = x <==> f(x)=( 1/g(x) )
int[g(x)]d[x] [o(x)o] ( int[f(x)]d[x]+f(x) ) = x <==> f(x)=e^{(-x)}·int[( 1/g(x) )·e^{x}]d[x]
int[g(x)]d[x] [o(x)o] ( int[f(x)]d[x]+d_{x}[f(x)] ) = x <==> ...
...f(x)=int[ cos(s)·int[( 1/g(x) )·cos(x)]d[x] ]d[x]+int[ sin(s)·int[( 1/g(x) )·sin(x)]d[x] ]d[x]


int[g(x)]d[x] [o(x)o] int[f(x)]d[x] = ax <==> f(x)=( a/g(x) )
int[g(x)]d[x] [o(x)o] ( int[f(x)]d[x]+f(x) ) = ax <==> f(x)=e^{(-x)}·int[( a/g(x) )·e^{x}]d[x]
int[g(x)]d[x] [o(x)o] ( int[f(x)]d[x]+d_{x}[f(x)] ) = ax <==> ...
...f(x)=int[ cos(s)·int[( a/g(x) )·cos(x)]d[x] ]d[x]+int[ sin(s)·int[( a/g(x) )·sin(x)]d[x] ]d[x]

dualogia

g(x)+h(y)=f(x)


x+y = 0 <==> y=(-x)
x+y=(x+(-a))^{n} <==> ( x=a <==> y=(-a) )
x+y=(x^{p}+(-a)) <==> ( x=a^{(1/p)} <==> y=(-1)a^{(1/p)} )
x+y=(e^{px}+(-a)) <==> ( x=ln(a^{(1/p)}) <==> y=(-1)ln(a^{(1/p)}) )
x+y=(ln(px)+(-a)) <==> ( x=e^{a+(-1)ln(p)} <==> y=(-1)e^{a+(-1)ln(p)}) )
x+y=( (a^{(k/p)}x^{p+(-k)})+(-a) ) <==> ( x=a^{(1/p)} <==> y=(-1)a^{(1/p)} )
x+y=(e^{p·x^{m}}+(-a)) <==> ( x=( ln(a^{(1/p)}) )^{(1/m)} <==> y=(-1)( ln(a^{(1/p)}) )^{(1/m)} )


x^{2}+y^{2} = 0 <==> y=ix or y=(-i)x
x^{n}+y^{n} = 0 <==> y=e^{(1/n)·pi·i}x or y=e^{(-1)(1/n)·pi·i}x


x^{n+1}+y^{n+1} = n^{n+1} <==> ( x=cos[n](t) <==> y=sin[n](t) )
x^{n+1}+y^{n+1} = 1 <==> ( x=(cos[n](t)/n) <==> y=(sin[n](t)/n) )






d_{x}[g(x)]+d_{y}[h(y)]d_{x}[y]=d_{x}[f(x)]


d_{y}[h(y)]d_{x}[y]=d_{x}[f(x)]+(-1)d_{x}[g(x)]
d_{y}[h(y)]d_{x}[y]=d_{x}[f(x)+(-1)g(x)]
d_{y}[h(y)]d_{x}[y]=d_{x}[h(y)]


int[g(x)]d[x]·d_{x}[y]+int[h(y)]d[y]=int[f(x)]d[x]·d_{x}[y]


int[h(y)]d[y]=( int[f(x)]d[x]+(-1)int[g(x)]d[x] )·d_{x}[y]
int[h(y)]d[y]=int[f(x)+(-1)g(x)]d[x]·d_{x}[y]
int[h(y)]d[y]=int[h(y)]d[x]·d_{x}[y]
int[h(y)]d[y]=int[h(y)]d[y]


F(x)=g(x)h(y)
F(x)=g(x)f(x)+(-1)( g(x) )^{2}
f(x)=( F(x)/g(x) )+g(x)
g(x)+h(y)=( F(x)/g(x) )+g(x)
h(y)=( F(x)/g(x) )
g(x)h(y)=F(x)

d_{x}[F(x)]=d_{x}[g(x)]f(x)+g(x)d_{x}[f(x)]+(-2)g(x)d_{x}[g(x)]
d_{x}[g(x)h(y)]=d_{x}[g(x)]f(x)+g(x)d_{x}[f(x)]+(-2)g(x)d_{x}[g(x)]
d_{x}[g(x)]h(y)+g(x)d_{x}[h(y)]=d_{x}[g(x)]( f(x)+(-1)g(x) )+g(x)d_{x}[f(x)+(-1)g(x)]
d_{x}[g(x)]h(y)+g(x)d_{x}[h(y)]=d_{x}[g(x)]h(y)+g(x)d_{x}[h(y)]


int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] int[h(y)]d[x]
int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] int[f(x)]d[x]+(-1)int[g(x)]d[x] [o(x)o] int[g(x)]d[x]
int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] ( int[f(x)]d[x]+(-1)int[g(x)]d[x] )
int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] ( int[f(x)+(-1)g(x)]d[x] )
int[F(x)]d[x] = int[g(x)]d[x] [o(x)o] int[h(y)]d[x]




x+y=0
int[y]d[y]=int[(-x)]d[(-x)]
int[y]d[y]=(1/2)(-x)^{2}
d_{x}[y]=d_{x}[(-x)]
d_{x}[y]=(-1)
F(x)=(-1)x^{2}


x+y=(x+(-a))^{n}
int[y]d[y]=int[(x+(-a))^{n}+(-x)]d[(x+(-a))^{n}+(-x)]
int[y]d[y]=(1/2)( (x+(-a))^{n}+(-x) )^{2}
d_{x}[y]=d_{x}[(x+(-a))^{n}+(-x)]
d_{x}[y]=n(x+(-a))^{(n+(-1))}+(-1)
F(x)=x(x+(-a))^{n}+(-1)x^{2}
F(a)=(-1)a^{2}


x+y=(x^{p}+(-a))
int[y]d[y]=int[(x^{p}+(-a))+(-x)]d[(x^{p}+(-a))+(-x)]
int[y]d[y]=(1/2)( (x^{p}+(-a))+(-x) )^{2}
d_{x}[y]=d_{x}[(x^{p}+(-a))+(-x)]
d_{x}[y]=px^{(p+(-1))}+(-1)
F(x)=(x^{p+1}+(-a)x)+(-1)x^{2}
F(a^{(1/p)})=(-1)( a^{(1/p)} )^{2}

aplicacions lineals

f(x,y) = <ax+by,cx+dy>


ker(f):
ax+by=0
cx+dy=0


x+(b/a)y=0
(c/d)x+y=0


x+(b/a)y=0
0·x+(1+(-1)(c/d)(b/a))y=0


Si ( a=b & c=d )  ==> x+y=0 ==> ( ( x=1 & y=(-1) ) or ( x=(-1) & y=1 ) )
Si ( ad=cb )  ==> x+(b/a)y=0 ==> ( ( x=1 & y=( (-a)/b ) ) or ( x=( (-b)/a) ) &  y=1 )
Si ( ad!=cb )  ==> ( x=0 & y=0 )
( ad=cb ) <==> ((-d)/c) = ((-b)/a)
Im(f):
ax+by=1
cx+dy=1
si ad!=cb ==> ( y=(a+(-c))/(ad+(-1)cb) & x=(d+(-b))/(ad+(-1)cb) )
ax+by=n
cx+dy=n
si ad!=cb ==> ( y=n·(a+(-c))/(ad+(-1)cb) & x=n·(d+(-b))/(ad+(-1)cb) )
ax+by=m
cx+dy=n
si ad!=cb ==> ( y=(an+(-c)m)/(ad+(-1)cb) & x=(dm+(-b)n)/(ad+(-1)cb) )


f(x,y) = <ax+by,cx+dy,ux+vy>


ker(f):
ax+by=0
cx+dy=0
ux+vy=0


x+(b/a)y=0
(c/d)x+y=0
(u/v)x+y=0


x+(b/a)y=0
0·x+(1+(-1)(c/d)(b/a))y=0
0·x+(1+(-1)(u/v)(b/a))y=0


Si ( a=b & c=d & u=v )  ==> x+y=0 ==> ( ( x=1 & y=(-1) ) or ( x=(-1) & y=1 ) )
Si ( ad=cb & av=ub)  ==> x+(b/a)y=0 ==> ( ( x=1 & y=( (-a)/b ) ) or ( x=( (-b)/a) ) &  y=1 )
Si ( ad!=cb or av!=ub )  ==> ( x=0 & y=0 )
( ad=cb & av=ub) <==> ( ((-d)/c) = ((-b)/a) & ((-v)/u) = ((-b)/a) )


Im(f):
ax+by=1+(-1)( (ud+(-1)cv)/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) )
cx+dy=1+(-1)( (bu+(-1)av)/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) )
ux+vy=1+(-1)( (ad+(-1)bc)/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) )
si ( ad!=cb or ud!=cv or av!=bu ) ==>...
... ( y=(a+(-c)+u)/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) &...
.... x=((-b)+d+(-v))/( (ad+(-1)bc)+(ud+(-1)cv)+(bu+(-1)av) ) )

aplicacions lineals

f(x,y) = ax+by


ker(f):
f(x,y)=0
ax+by=0
<x,y>=k·<(-b),a>=(-k)·<b,(-a)>=ak·<(-b)/a,1>
<x,y>=k·<(-b)/a,1>
<x,y>=k·<b,(-a)>=(-k)·<(-b),a>=bk·<1,(-a)/b>
<x,y>=k·<1,(-a)/b>


Im(f):
f(x,y)=v
ax+by=v
<x,y>=(v/2)·<(1/a),(1/b)>


f(x,y,z) = ax+by+cz


ker(f):
f(x,y,z)=0
ax+by+cz=0
<x,y,z>=k·<(-c),(-c),(a+b)>=(-k)·<c,c,(-1)(a+b)>=(-c)k·<(1,0,(-a)/c>+(-c)k·<(0,1,(-b)/c>
si ( x=1 & y=0 ) ==> <x,y,z>=k·<(1,0,(-a)/c>
si ( x=0 & y=1 ) ==> <x,y,z>=k·<(0,1,(-b)/c>
<x,y,z>=k·<(-b),(a+c),(-b)>=(-k)·<b,(-1)(a+c),b>=(-b)k·<(1,(-a)/b,0>+(-b)k·<(0,(-c)/b,1>
si ( x=1 & z=0 ) ==> <x,y,z>=k·<(1,(-a)/b,0>
si ( x=0 & z=1 ) ==> <x,y,z>=k·<(0,(-c)/b,1>
<x,y,z>=k·<(b+c),(-a),(-a)>=(-k)·<(-1)(b+c),a,a>=(-a)k·<((-c)/a),0,1>+(-a)k·<((-b)/a),1,0>
si ( y=0 & z=1 ) ==> <x,y,z>=k·<((-c)/a),0,1>
si ( y=1 & z=0 ) ==> <x,y,z>=k·<((-b)/a),1,0>


k·<(1,(-a)/b,0>=k·<(1,0,(-a)/c>+((-a)/b)k·<(0,1,(-b)/c>
k·<(0,(-c)/b,1>=0·k·<(1,0,(-a)/c>+((-c)/b)k·<(0,1,(-b)/c>
k·<((-c)/a),0,1>=((-c)/a)·k·<(1,0,(-a)/c>+0·k·<(0,1,(-b)/c>
k·<((-b)/a),1,0>=((-b)/a)·k·<(1,0,(-a)/c>+k·<(0,1,(-b)/c>


k·<(1,0,(-a)/c>=k·<(1,(-a)/b,0>+((-a)/c)k·<(0,(-c)/b,1>
k·<(0,1,(-b)/c>=0·k·<(1,(-a)/b,0>+((-b)/c)k·<(0,(-c)/b,1>
k·<((-c)/a),0,1>=((-c)/a)k·<(1,(-a)/b,0>+k·<(0,(-c)/b,1>
k·<((-b)/a),1,0>=((-b)/a)k·<(1,(-a)/b,0>+0·k·<(0,(-c)/b,1>


k·<(1,0,(-a)/c>=((-a)/c)k·<((-c)/a,0,1>+0·k·<((-b)/a,1,0>
k·<(0,1,(-b)/c>=((-b)/c)k·<((-c)/a,0,1>+k·<((-b)/a,1,0>
k·<(1,(-a)/b,0>=0·k·<((-c)/a,0,1>+((-a)/b)k·<((-b)/a,1,0>
k·<(0,(-c)/b),1>=k·<((-c)/a,0,1>+((-c)/b)k·<((-b)/a,1,0>






dim( ker(f) )=2


Im(f):
f(x,y,z)=v
ax+by+cz=v
<x,y,z>=(v/3)·<(1/a),(1/b),(1/c)>

polinomis y espai vectorial quocient

E = s·(x^{2}+1)+i·(x+1)+j·1
F = k·(x^{2}+x+1)


s·(x^{2}+1)+i·(x+1)+j·1+(-k)·(x^{2}+x+1) € [E]_{F}


(p+1)k·(x^{2}+1)+(p+1)k·(x+1)+((-p)+(-1))k·1+(-k)·(x^{2}+x+1) = pk·(x^{2}+x+1)
(p+1)k·(x^{2}+1)+(p+1)k·(x+1)+((-p)+(-1))k·1 = [ pk·(x^{2}+x+1) ]+k·(x^{2}+x+1)
[ pk·(x^{2}+x+1) ]+k·(x^{2}+x+1) = (p+1)k·(x^{2}+x+1)


(m+1)k·(x^{2}+1)+(n+1)k·(x+1)+(q+(-1))k·1+(-k)·(x^{2}+x+1) = mk·(x^{2}+1)+nk·(x+1)+qk·1
(m+1)k·(x^{2}+1)+(n+1)k·(x+1)+(q+(-1))k·1 = [ mk·(x^{2}+1)+nk·(x+1)+qk·1 ]+k·(x^{2}+x+1)


dim(E/F) = dim( [E]_{F} )+dim(F) =4
[0]_{F} = pk·(x^{2}+x+1)
[u]_{F} = mk·(x^{2}+1)+nk·(x+1)+qk·1
F=[0]_{F}+F
E/F=[u]_{F}+F


E = s·x^{2}+i·(x+1)+j·1
F = k·(x^{2}+x+1)


s·x^{2}+i·(x+1)+j·1 + (-k)·(x^{2}+x+1) € [E]_{F}




(p+1)k·x^{2}+(p+1)k·(x+1)+0·1 + (-k)·(x^{2}+x+1) = pk·(x^{2}+x+1)
(p+1)k·x^{2}+(p+1)k·(x+1)+0·1= [ pk·(x^{2}+x+1) ]+k·(x^{2}+x+1)
[ pk·(x^{2}+x+1) ]+k·(x^{2}+x+1) = (p+1)k·(x^{2}+x+1)


(m+1)k·x^{2}+(n+1)k·(x+1)+q·1 + (-k)·(x^{2}+x+1) = mk·x^{2}+nk·(x+1)+q·1
(m+1)k·x^{2}+(n+1)k·(x+1)+q·1 = [ mk·x^{2}+nk·(x+1)+q·1 ]+k·(x^{2}+x+1)




dim(E/F)=dim( [E]_{F} )+dim(F)=4
[0]_{F} = pk·(x^{2}+x+1)
[u]_{F} = mk·x^{2}+nk·(x+1)+q·1
F=[0]_{F}+F
E/F=[u]_{F}+F

espai vectorial quocient

E = i·<1,0>+j·<0,1>
F = k·<1,1>


i·<1,0>+j·<0,1>+(-k)·<1,1> € [E]_{F}


(p+1)k·<1,0>+(p+1)k·<0,1>+(-k)·<1,1> = pk·<1,1>
(p+1)k·<1,0>+(p+1)k·<0,1> = [ pk·<1,1> ]+k·<1,1>
[ pk·<1,1> ]+k·<1,1> = (p+1)k·<1,1>


(m+1)k·<1,0>+(n+1)k·<0,1>+(-k)·<1,1> = mk·<1,0>+nk·<0,1>
(m+1)k·<1,0>+(n+1)k·<0,1> = [ mk·<1,0>+nk·<0,1> ]+k·<1,1>


dim(E/F) = dim( [E]_{F} )+dim(F) =3
[0]_{F} = pk·<1,1>
[u]_{F} = mk·<1,0>+nk·<0,1>
F=[0]_{F}+F
E/F=[u]_{F}+F

polinomis y espai vectorial suma y intersecció

E = k·(x^{2}+x+1)
F = i·(x+1)+j·1


La suma
E+F = k·x^{2}+(k+i)·x+(k+i+j)·1
si k=0 ==>  E+F = i·(x+1)+j·1
si ( i=0 & j=0 ) ==>  E+F = k·(x^{2}+x+1)


La intersecció
k·(x^{2}+x+1) = i·(x+1)+j·1
0·(x^{2}+x+1) = 0·(x+1)+0·1
E[M]F = {0}




E = k·(x^{2}+x+1)
F = s·x^{2}+i·(x+1)+j·1


La suma
E+F = (k+s)·x^{2}+(k+i)·x+(k+i+j)·1
si k=0 ==>  E+F = s·x^{2}+i·(x+1)+j·1
si ( s=0 & i=0 & j=0 ) ==>  E+F = k·(x^{2}+x+1)


La intersecció
k·(x^{2}+x+1) = s·x^{2}+i·(x+1)+j·1
k·(x^{2}+x+1) = k·x^{2}+k·(x+1)+0·1
k·(x^{2}+x+1) = k·(x^{2}+x+1)
E[M]F = k·(x^{2}+x+1)


E = k·x
F = i·(x+1)+j·1


La suma
E+F = (k+i)·x+(i+j)·1
si k=0 ==>  E+F = i·(x+1)+j·1
si ( i=0 & j=0 ) ==>  E+F = k·x


La intersecció
k·x = i·(x+1)+j·1
k·x = k·(x+1)+(-k)·1
k·x = k·x
E[M]F = k·x




dim(E+F) = dim(E)+dim(F)+(-1)dim(E[M]F)

serie zeta alternada

teorema:
∑ ( (-1)^{n}/n^{2s} ) =  (-1)^{s}·pi^{2·s}·( 2^{2·s+(-1)}+(-1) )·B_{2s+(-1)}(0)


∑ ( (-1)^{n}/n^{2} ) =  (-1)·(pi^{2}/12)
∑ ( (-1)^{n}/n^{2} ) =  (-1)^{1}·pi^{2·1}·(2^{2·1+(-1)}+(-1))·(1/12)
∑ ( (-1)^{n}/n^{4} ) =  (-1)·(7pi^{4}/720)
∑ ( (-1)^{n}/n^{4} ) =  (-1)^{2}·pi^{2·2}·(2^{2·2+(-1)}+(-1))·(-1)(1/720)
∑ ( (-1)^{n}/n^{6} ) =  (-1)·(31pi^{6}/30240)
∑ ( (-1)^{n}/n^{6} ) =  (-1)^{3}·pi^{2·3}·(2^{2·3+(-1)}+(-1))·(1/30240)
∑ ( (-1)^{n}/n^{8} ) = (-1)(127pi^{8}/1209600)
∑ ( (-1)^{n}/n^{8} ) =  (-1)^{4}·pi^{2·4}·( 2^{2·4+(-1)}+(-1) )·(-1)(1/1209600)
∑ ( (-1)^{n}/n^{10} ) = (-1)(511pi^{10}/47900160)
∑ ( (-1)^{n}/n^{10} ) =  (-1)^{5}·pi^{2·5}·( 2^{2·5+(-1)}+(-1) )·(1/47900160)




teorema:
∑ ( (-1)^{n+1}/n^{2s} ) =  ( 1+( (-1)/2^{2s+(-1)} ) )·∑ ( 1/n^{2s} )




demostració:
(-1)∑ ( (-1)^{n}/n^{2s} ) =  (-1)^{s+1}·pi^{2·s}·( 2^{2·s+(-1)}+(-1) )·B_{2s+(-1)}(0)


(-1)∑ ( (-1)^{n}/n^{2s} ) =  (-1)^{s+1}·pi^{2·s}·( 2^{2·s+(-1)}·B_{2s+(-1)}(0)+...
...+(-1)^{s}·pi^{2·s}B_{2s+(-1)}


(-1)∑ ( (-1)^{n}/n^{2s} ) =  ∑ ( 1/n^{2s} )+(-1)^{s}·pi^{2·s}B_{2s+(-1)}


(-1)∑ ( (-1)^{n}/n^{2s} ) =  ∑ ( 1/n^{2s} )+...
...+( (-1)/2^{2s+(-1)} )·(-1)^{s+1}·pi^{2·s}·2^{2s+(-1)}B_{2s+(-1)}




(-1)∑ ( (-1)^{n}/n^{2s} ) =  ∑ ( 1/n^{2s} )+( (-1)/2^{2s+(-1)} )·∑ ( 1/n^{2s} )


(-1)∑ ( (-1)^{n}/n^{2s} ) =  ( 1+( (-1)/2^{2s+(-1)} ) )·∑ ( 1/n^{2s} )

coment

not six zero energy-drink <==> cubata-energético de 40 grados.
four wheeles motor-car <==> motor-carro de 4 ruedas.
tw wheeles motor-ciclet <==> motor-cicleta de 2 ruedas.
four cuorter pizza <==> 4 cuartos de pizza