求解一道关于极限的题目f(x)=x^2+x,a(n+1)=f(an),a1=0.5求lim[1/(1+a1)+1/(1+a2)+.+1/(1+an)] (n趋向于无穷){an}为一数列
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![求解一道关于极限的题目f(x)=x^2+x,a(n+1)=f(an),a1=0.5求lim[1/(1+a1)+1/(1+a2)+.+1/(1+an)] (n趋向于无穷){an}为一数列](/uploads/image/z/3300413-5-3.jpg?t=%E6%B1%82%E8%A7%A3%E4%B8%80%E9%81%93%E5%85%B3%E4%BA%8E%E6%9E%81%E9%99%90%E7%9A%84%E9%A2%98%E7%9B%AEf%28x%29%3Dx%5E2%2Bx%2Ca%28n%2B1%29%3Df%28an%29%2Ca1%3D0.5%E6%B1%82lim%5B1%2F%281%2Ba1%29%2B1%2F%281%2Ba2%29%2B.%2B1%2F%281%2Ban%29%5D+%28n%E8%B6%8B%E5%90%91%E4%BA%8E%E6%97%A0%E7%A9%B7%EF%BC%89%7Ban%7D%E4%B8%BA%E4%B8%80%E6%95%B0%E5%88%97)
求解一道关于极限的题目f(x)=x^2+x,a(n+1)=f(an),a1=0.5求lim[1/(1+a1)+1/(1+a2)+.+1/(1+an)] (n趋向于无穷){an}为一数列
求解一道关于极限的题目
f(x)=x^2+x,a(n+1)=f(an),a1=0.5
求lim[1/(1+a1)+1/(1+a2)+.+1/(1+an)] (n趋向于无穷)
{an}为一数列
求解一道关于极限的题目f(x)=x^2+x,a(n+1)=f(an),a1=0.5求lim[1/(1+a1)+1/(1+a2)+.+1/(1+an)] (n趋向于无穷){an}为一数列
a(n+1)=f(an)=an^2+an
1/a(n+1)=1/(an^2+an)=1/[an(1+an)]=1/an - 1/(1+an)
1/(1+an)=1/an-1/a(n+1)
1/(1+a1)+1/(1+a2)+.+1/(1+an)
=(1/a1-1/a2)+(1/a2-1/a3)+...[1/an-1/a(n+1)]
=1/a1-1/a(n+1)
=2-1/a(n+1)
a(n+1)=an^2+an
a(n+1)-an=an^2>0 (a1>0)
a(n+1)>an>a(n-1)>...>a1=1/2
a2-a1=a1^2
a3-a2=a2^2
.
an-a(n-1)=a(n-1)^2(上述式子进行叠加)
an-a1=a1^2+a2^2+a3^2+...+a(n-1)^2
an=a1+a1^2+a2^2+a3^2+...+a(n-1)^2
>1/2+(1/2)^2+(1/2)^2+(1/2)^2+...+(1/2)^2 (共有n-1个(1/2)^2 )
=1/2+(n-1)/4
=(n+1)/4
a(n+1)>[(n+1)+1]/4=(n+2)/4
1/a(n+1)