problemes de números

s=1+(1/2)+...+(1/n)  ==> ¬( s€N ) or s=1


s = sum[k=1-->n]( (1·...·(k+(-1))(k+1)·...·n)/n! )
sigui m_{k}€N ==>
s = sum[k=1-->n]( m_{k} )=sum[k=1-->n]( (1·...·(k+(-1))(k+1)·...·n)/n! )
sum[k=1-->n]( m_{k} )n!=sum[k=1-->n]( (1·...·(k+(-1))(k+1)·...·n) )
m_{k}=(1/k)


s=1+(1/3)+...+(1/(2n+1))  ==> ¬( s€N ) or s=1


s = sum[k=1-->n]( (1·...·(2(k+(-1))+1)(2(k+1)+1)·...·(2n+1))/(2n+1)!! )
sigui m_{k}€N ==>
s = sum[k=1-->n]( m_{k} )=sum[k=1-->n]( (1·...·(2(k+(-1))+1)(2(k+1)+1)·...·(2n+1))/(2n+1)!! )
sum[k=1-->n]( m_{k} )(2n+1)!!=sum[k=1-->n]( (1·...·(2(k+(-1))+1)(2(k+1)+1)·...·(2n+1)) )
m_{k}=(1/(2k+1))


s=1+(1/2!)+...+(1/n!)  ==> ¬( s€N ) or s=1


s = sum[k=1-->n]( (1·...·(k+(-1))!(k+1)!)·...·n!)/(n!)! )
sigui m_{k}€N ==>
s = sum[k=1-->n]( m_{k} )=sum[k=1-->n]( (1·...·(k+(-1))!(k+1)!)·...·n!)/(n!)! )
sum[k=1-->n]( m_{k} )(n!)!=sum[k=1-->n]( (1·...·(k+(-1))!(k+1)!)·...·n!) )
m_{k}=(1/k!)


s=(p/n)+...(n)...+(p/n)==> s€N


s = ( (p+...(n)...+p)/n )
sigui m_{k}€N ==>
s = sum[k=1-->p]( m_{k} )=( ( p+...(n)...+p )/n )
( m_{1}+...(p)...+m_{p} )=(np/n)
( 1+...(p)...+1 )=p
m_{k}=1


s=(p/n^{q})+...(n)...+(p/n^{q})==> ¬( s€N ) or p=n^{q+(-1)}


s = ( (p+...(n)...+p)/n^{q} )
sigui m_{k}€N ==>
s = sum[k=1-->p]( m_{k} )=( ( p+...(n)...+p )/n^{q} )
( m_{1}+...(p)...+m_{p} )=(np/n^{q})
( 1/n^{q+(-1)}+...(p)...+1/n^{q+(-1)} )n^{q+(-1)}=p
m_{k}=(1/n^{q+(-1)})
( 1/n^{q+(-1)}+...(n^{q+(-1)})...+1/n^{q+(-1)} )n^{q+(-1)}=n^{q+(-1)}