sumas harmónicas impares

Sum[ 1/n^{3} ] = pi^{3}·f(3)·( 1/(2^{2}+1) )·( 1/(2^{1}+(-1)) ) = ( pi^{3}/20 )

f(3) = (1/4) & 4 = 1·1·(6+(-2))

Sum[ 1/n^{5} ] = pi^{5}·f(5)·( 1/(2^{3}+1) )·( 1/(2^{2}+(-3)) ) = ( pi^{5}/288 )

f(5) = (1/32) & 32 = 2·2·(10+(-2))

Sum[ 1/n^{7} ] = pi^{7}·f(7)·( 1/(2^{3}+3) )·( 1/(2^{2}+(-1)) ) = ( pi^{7}/2970 )

f(7) = (1/90) & 90 = 3·3·(14+(-4))

Sum[ 1/n^{9} ] = pi^{9}·f(9)·( 1/(2^{4}+1) )·( 1/(2^{3}+(-7)) ) = ( pi^{9}/29750 )

f(9) = (1/1750) & 1750 = 5·5·5·(18+(-4))

... (pi^{10}/93555) < (pi^{9}/29750) < ... 

... (pi^{8}/9450) < (pi^{7}/2970) < ...

... (pi^{6}/945) < (pi^{5}/288) < ...

... (pi^{4}/90) < (pi^{3}/20) < ...

... (pi^{2}/6)

funció zeta 3s

∑ ( 1/k^{3(2s+1)} ) = (1/2)·( 1/( (2^{2s}+1)^{2s+1}·( 2^{2s+1}+(-1) )·( 2^{2s+2}+1 ) ) )·pi^{3(2s+1)}


∑ ( 1/k^{3} ) = ( 1/(2·2^{1}·5) )·pi^{3} = (1/20)·pi^{3}
∑ ( 1/k^{9} ) = ( 1/(2·5^{3}·7·17) )·pi^{9} = (1/29750)·pi^{9}


(1/20)·pi^{3} < (1/6)·pi^{2}
pi < (20/6) = 3.3...


(1/90)·pi^{4} < (1/20)·pi^{3}
pi < (90/20) = 4.5


(1/29750)·pi^{9} < (1/9450)·pi^{8}
pi < (29750/9450) = 3.1481...


(1/93555)·pi^{10} < (1/29750)·pi^{9}
pi < (93555/29750) = 3.1447058824

índex de matemàtiques

index de etiquetes matemàtiques

serie geométrica

∑ ( x^{k} ) = ( (x^{n+1}+(-1))/(x+(-1)) )


Si 0 < x < 1 ==> ∑ ( x^{k} ) < ( 1/(1+(-x)) )


∑ ( x^{pk} ) = ( (x^{p(n+1)}+(-1))/(x^{p}+(-1)) )


Si 0 < x < 1 ==> ∑ ( x^{pk} ) < ( 1/(1+(-1)·x^{p}) )


∑ ( e^{kx} ) = ( (e^{(n+1)x}+(-1))/(e^{x}+(-1)) )


Si x < 0 ==> ∑ ( e^{kx} ) < ( 1/(1+(-1)·e^{x}) )

series numériques telescópiques

∑ ( 1/(n(n+1)) ) = ∑ ( ( 1/n )+(-1)( 1/(n+1) ) ) < 1


∑ ( m/(n(n+m)) ) = ∑ ( ( 1/n )+(-1)( 1/(n+m) ) ) < 1+(1/2)+...+(1/m)


∑ ( 1/(n(n+m)) ) < (1/m)·( 1+(1/2)+...+(1/m) )


∑ ( 2/(n(n+2)) ) = ∑ ( ( 1/n )+(-1)( 1/(n+2) ) ) < (3/2)
∑ ( 3/(n(n+3)) ) = ∑ ( ( 1/n )+(-1)( 1/(n+3) ) ) < (11/6)


∑ ( 1/(n(n+2)) ) < (3/4)
∑ ( 1/(n(n+3)) ) < (11/18)

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} )

series de Fourier

x^{2} = (pi^{2}/3)+4·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{2} ) = ( pi^{2}/6 )
si x=0 ==> ∑ ( (-1)^{k}/k^{2} ) = (-1)( pi^{2}/12 )


x^{4} = (pi^{4}/5)+8pi^{2}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+..
...+(-48)·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{4} ) = ( pi^{4}/90 )
si x=0 ==> ∑ ( (-1)^{k}/k^{4} ) = (-1)( 7pi^{4}/720 )


x^{6} = (pi^{6}/7)+12pi^{4}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+...
...+(-240)pi^{2}·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )+...
...+1440·∑ ( ( 1/k^{6} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{6} ) = ( pi^{6}/945 )
si x=0 ==> ∑ ( (-1)^{k}/k^{6} ) = (-1)( 31pi^{4}/30240 )


x^{8} = (pi^{8}/9)+16pi^{6}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+...
...+(-672)pi^{4}·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )+...
...+13440pi^{2}·∑ ( ( 1/k^{6} )·cos(k·pi)·cos(kx) )+...
...+(-80640)·∑ ( ( 1/k^{8} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{8} ) = ( pi^{8}/9450 )


x^{10} = (pi^{10}/11)+20pi^{8}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+...
...+(-1440)pi^{6}·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )+...
...+60480pi^{4}·∑ ( ( 1/k^{6} )·cos(k·pi)·cos(kx) )+...
...+(-1209600)pi^{2}·∑ ( ( 1/k^{8} )·cos(k·pi)·cos(kx) )+...
...+7257600·∑ ( ( 1/k^{10} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{10} ) = ( pi^{10}/93555 )

polinomis de suma de potencies

B_{0}(x) = x+(-1)(1/2)
B_{1}(x) = (1/2)x^{2}+(-1)(1/2)x+(1/12)
B_{2}(x) = (1/6)x^{3}+(-1)(1/4)x^{2}+(1/12)x
B_{3}(x) = (1/24)x^{4}+(-1)(1/12)x^{3}+(1/24)x^{2}+(-1)(1/720)
B_{4}(x) = (1/120)x^{5}+(-1)(1/48)x^{4}+(1/72)x^{3}+(-1)(1/720)x
B_{5}(x) = (1/720)x^{6}+(-1)(1/240)x^{5}+(1/288)x^{4}+(-1)(1/1440)x^{2}+...
...+(1/30240)
B_{6}(x) = (1/5040)x^{7}+(-1)(1/1440)x^{6}+(1/1440)x^{5}+(-1)(1/4320)x^{3}+...
...+(1/30240)x
B_{7}(x) = (1/40320)x^{8}+(-1)(1/10080)x^{7}+(1/8640)x^{6}+(-1)(1/17280)x^{4}+...
...+(1/60480)x^{2}+(-1)(1/1209600)
B_{8}(x) = (1/362880)x^{9}+(-1)(1/80640)x^{8}+(1/60480)x^{7}+(-1)(1/86400)x^{5}+...
...+(1/181440)x^{3}+(-1)(1/1209600)x
B_{9}(x) = (1/3628800)x^{10}+(-1)(1/725760)x^{9}+(1/483840)x^{8}+(-1)(1/518400)x^{6}+...
...+(1/725760)x^{4}+(-1)(1/2419200)x^{2}+(1/47900160)
B_{10}(x) = (1/39916800)x^{11}+(-1)(1/7257600)x^{10}+(1/4354560)x^{9}+(-1)(1/3628800)x^{7}+...
...+(1/3628800)x^{5}+(-1)(1/7257600)x^{3}+(1/47900160)x+B_{10}(0)


B_{n+1}(1) = B_{n+1}(0)
d_{x}[B_{n+1}(x)] = B_{n}(x)


1^{m}+...+n^{m} = m!( B_{m}(n+1)+(-1)B_{m}(1) )

sumes de potencies

1^{0}+...+n^{0} = n
1^{1}+...+n^{1} = (1/2)n(n+1)
1^{2}+...+n^{2} = (1/6)n(n+1)(2n+1)
1^{3}+...+n^{3} = (1/4)n^{2}(n+1)^{2}
1^{4}+...+n^{4} = (1/30)n(n+1)(2n+1)(3n^{2}+3n+(-1))
1^{5}+...+n^{5} = (1/12)n^{2}(n+1)^{2}(2n^{2}+2n+(-1))
1^{6}+...+n^{6} = (1/42)n(n+1)(2n+1)(3n^{4}+6n^{3}+(-3)n+1)
1^{7}+...+n^{7} = (1/24)n^{2}(n+1)^{2}(3n^{4}+6n^{3}+(-1)n^{2}+(-4)n+2)
1^{8}+...+n^{8} = (1/90)n(n+1)(2n+1)(n^{2}+n+(-1))(5n^{4}+10n^{3}+(-5)n+4)+...
                            ...+(1/90)n(n+1)(2n+1)
1^{9}+...+n^{9} = (1/20)n^{2}(n+1)^{2}(n^{2}+n+(-1))(2n^{4}+4n^{3}+(-n^{2})+(-3)n+3)


he arribat amb polinomis de sumes de potencies fins a n^{9} y metode del binomi para sumes de potencies fins a n^{5}


método del binomi para sumes de potencies:


(n+1)^{n}+(-1)=[ n // 1 ]·∑ ( k^{n+(-1)} )+...+[ n // n+(-1) ]·∑ ( k )+n


teorema:
si (1/oo^{n}) = 0 ==> lim[n-->oo]( (1/n^{m+1})·( 1^{m}+...+n^{m} ) ) = ( 1/(m+1) )

serie zeta

∑ (1/n^{2}) =  (pi^{2}/6)
∑ (1/n^{2}) =  (-1)^{1+1}·pi^{2·1}·2^{2·1+(-1)}·(1/12)
∑ (1/n^{4}) = (pi^{4}/90)
∑ (1/n^{4}) =  (-1)^{2+1}·pi^{2·2}·2^{2·2+(-1)}·(-1)(1/720)
∑ (1/n^{6}) = (pi^{6}/945)
∑ (1/n^{6}) =  (-1)^{3+1}·pi^{2·3}·2^{2·3+(-1)}·(1/30240)
∑ (1/n^{8}) = (pi^{8}/9450)
∑ (1/n^{8}) =  (-1)^{4+1}·pi^{2·4}·2^{2·4+(-1)}·(-1)(1/1209600)
∑ (1/n^{10}) = (pi^{10}/93555)
∑ (1/n^{10}) =  (-1)^{5+1}·pi^{2·5}·2^{2·5+(-1)}·(1/47900160)


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


on B_{2s+(-1)}(0) és el número de suma de potencies.