a1=0,a2=1,an=(an-1+an-2)/2 求证数列是收敛的,并求极限.2楼3楼都不对2楼看错了题3楼,a4>a5不能用单调性
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![a1=0,a2=1,an=(an-1+an-2)/2 求证数列是收敛的,并求极限.2楼3楼都不对2楼看错了题3楼,a4>a5不能用单调性](/uploads/image/z/5227860-12-0.jpg?t=a1%3D0%2Ca2%3D1%2Can%3D%28an-1%2Ban-2%29%2F2+%E6%B1%82%E8%AF%81%E6%95%B0%E5%88%97%E6%98%AF%E6%94%B6%E6%95%9B%E7%9A%84%2C%E5%B9%B6%E6%B1%82%E6%9E%81%E9%99%90.2%E6%A5%BC3%E6%A5%BC%E9%83%BD%E4%B8%8D%E5%AF%B92%E6%A5%BC%E7%9C%8B%E9%94%99%E4%BA%86%E9%A2%983%E6%A5%BC%EF%BC%8Ca4%3Ea5%E4%B8%8D%E8%83%BD%E7%94%A8%E5%8D%95%E8%B0%83%E6%80%A7)
a1=0,a2=1,an=(an-1+an-2)/2 求证数列是收敛的,并求极限.2楼3楼都不对2楼看错了题3楼,a4>a5不能用单调性
a1=0,a2=1,an=(an-1+an-2)/2 求证数列是收敛的,并求极限.
2楼3楼都不对
2楼看错了题
3楼,a4>a5不能用单调性
a1=0,a2=1,an=(an-1+an-2)/2 求证数列是收敛的,并求极限.2楼3楼都不对2楼看错了题3楼,a4>a5不能用单调性
这个先用特征方程求出an来,求法如下
an=(an-1+an-2)/2 的特征方程是x^2=(x+1)/2解出来的x1=-1/2,x2=2代入下式
an=c1*x1^n+c2*x2^n=c1*x1^n-1+(x2*c2)*x2^n-1
得an=c1*(-1/2)^n+c2(1)^n,再有a1,a2解出c1,c2.
最后得数列的通项公式是an=4/3*(-1/2)^n+2/3(有通项证明就应该不难了吧)
取极限的该数列的极限是2/3
a3= (a1+a2)/2 = 1/2
a4= (a3+a2)/2 = 3/4
an is bounded by 1
an is increasing
an 是收敛的
lim(n->∞)an
=1
a(n)-a(n-1) = (-1/2)·[a(n-1)-a(n-2)]
lim [a(n)-a(n-1)] = 0, a(n)收敛
a(n)-a(n-1) = (-1/2)^(n-2)·[a2-a1] = (-1/2)^(n-2)
a(n) = a(1)+ (-1/2)^0 + (-1/2)^1 +...+ (-1/2)^(n-2)
lim a(n) = lim a(1) + lim [(-1/2)^0 + (-1/2)^1 +...+ (-1/2)^(n-2)]
= 0 + 1/[1-(-1/2)] = 2/3
a3= (a1+a2)/2 = 1/2
a4= (a3+a2)/2 = 3/4
an is bounded by 1
an is increasing
an 是收敛的
lim(n->∞)an
=1 #