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.

conjugeited eit

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allí-nisteit <==> allá <==> allí-nek <==> allí.
aishí-nisteit <==> aishí <==> aishí-nek <==> así.

conjugeited eit

present
calcul-eitu <==> calculû <==> calculû-tek <==> calculo <==> calculo
calcul-eites <==> calcules <==> calcules-tek <==> calculas <==> calculai
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-eiting <==> -ant <==> -antu-dut <==> -ando


perfect
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