Sum
Moderatorer: Vektormannen, espen180, Aleks855, Solar Plexsus, Gustav, Nebuchadnezzar, Janhaa
De viktige leddene er
lg 1 = 0
lg 2 = 1
lg 4 = 2
lg 8 = 3
lg 16 = 4
lg 32 = 5
lg 64 = 6
lg 128 = 7
lg 256 = 8
lg 512 = 9
lg 1024 = 10
Summen blir derfor 10 + 9* (1023-511) + 8*(511-255)+7*(255-127)+6*(127-63)+5*(63-31)+4*(31-15)+3*(15-7)+2*(7-3)+(3-1)
Ser at den kan forenkles endel. F.eks. er -9*511+8*511=-511 etc.
Så summen forenkles til
10+9*1023-511-255-127-53-31-15-7-3-1
lg 1 = 0
lg 2 = 1
lg 4 = 2
lg 8 = 3
lg 16 = 4
lg 32 = 5
lg 64 = 6
lg 128 = 7
lg 256 = 8
lg 512 = 9
lg 1024 = 10
Summen blir derfor 10 + 9* (1023-511) + 8*(511-255)+7*(255-127)+6*(127-63)+5*(63-31)+4*(31-15)+3*(15-7)+2*(7-3)+(3-1)
Ser at den kan forenkles endel. F.eks. er -9*511+8*511=-511 etc.
Så summen forenkles til
10+9*1023-511-255-127-53-31-15-7-3-1
[tex]1024=2^10\,,\, 512=2^9\,,\, 256=2^8\,,\, 128=2^7 \, osv.[/tex]
Vi får
[tex]\sum_{N=1}^{1024} \lfloor \log_2\,N\rfloor=0+(4-2)+2(8-4)+3(16-8)+...+9(1024-512)+10 \\ \sum_{N=1}^{1024} \lfloor \log_2\,N\rfloor=10+\sum_{n=0}^{9}n(2^{n+1}-2^n)=8204[/tex]
Vi får
[tex]\sum_{N=1}^{1024} \lfloor \log_2\,N\rfloor=0+(4-2)+2(8-4)+3(16-8)+...+9(1024-512)+10 \\ \sum_{N=1}^{1024} \lfloor \log_2\,N\rfloor=10+\sum_{n=0}^{9}n(2^{n+1}-2^n)=8204[/tex]