尔雅{sin(n)/2n}数列不收敛。()
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(1)an=n^2cos2πn/3
cos2πn/3取到的值为-1/2,-1/2,1,-1/2,-1/2,1,.
对于n=3k(k∈N*),a(3k-2)+a(3k-1)+a(3k)=-1/2(3k-2)^2-1/2(3k-1)^2+(3k)^2=9k-5/2
所以S(3k)为{9k-5/2}的前k项和
S(3k)=9k(k+1)/2-5/2k=9/2k^2+2k
即当n=3k时,Sn=1/2n^2+2/3n
n=3k+1时,Sn=S(n-1)+an=1/2(n-1)^2+2/3(n-1)-1/2n^2=-1/3n-7/6
n=3k+2时,Sn=S(n+1)-a(n+1)=1/2(n+1)^2+2/3(n+1)-(n+1)^2=-1/2n^2-1/3n+1/6
cos2πn/3取到的值为-1/2,-1/2,1,-1/2,-1/2,1,.
对于n=3k(k∈N*),a(3k-2)+a(3k-1)+a(3k)=-1/2(3k-2)^2-1/2(3k-1)^2+(3k)^2=9k-5/2
所以S(3k)为{9k-5/2}的前k项和
S(3k)=9k(k+1)/2-5/2k=9/2k^2+2k
即当n=3k时,Sn=1/2n^2+2/3n
n=3k+1时,Sn=S(n-1)+an=1/2(n-1)^2+2/3(n-1)-1/2n^2=-1/3n-7/6
n=3k+2时,Sn=S(n+1)-a(n+1)=1/2(n+1)^2+2/3(n+1)-(n+1)^2=-1/2n^2-1/3n+1/6
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