大学里的数列 数列 A(N) A(1) = 2000 递推公式为:A(N+1) = arctan A(N)如何证明A(N)收敛于 PI/2
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大学里的数列 数列 A(N) A(1) = 2000 递推公式为:A(N+1) = arctan A(N)如何证明A(N)收敛于 PI/2
大学里的数列
数列 A(N) A(1) = 2000 递推公式为:A(N+1) = arctan A(N)
如何证明A(N)收敛于 PI/2
大学里的数列 数列 A(N) A(1) = 2000 递推公式为:A(N+1) = arctan A(N)如何证明A(N)收敛于 PI/2
A(N) 收敛于 0.
= = = = = = = = =
证明:因为 A(N+1) =arctan A(N),N=1,2,...
且 A(1) =2000>0,
所以 A(2) =arctan A(1) ∈(0,π/2),
A(3) =arctan A(2) ∈(0,π/2),
...
A(N+1) =arctan A(N) ∈(0,π/2).
所以 A(N) arctan A(N) = A(N+1),N=2,3,...
又因为 A(1) >A(2),A(N)>0,
所以 {A(N)} 是 单调递减 有下界.
所以 lim (n→∞) A(N) 存在,记为a,且 a∈[ 0,π/2),
则 lim (n→∞) A(N+1) =lim (n→∞) arctan A(N),
即 a =arctan a,a∈[ 0,π/2),
所以 a =0,
即 lim (n→∞) A(N) =0.
= = = = = = = = =
计算器验证:
(1) 角度制:
A(1) =2000,
A(2) =89.97135211
A(3) =89.36320374
A(4) =89.35887048
A(5) =89.35883939
A(6) =89.35883917
A(7) =89.35883917
.
(2) 弧度制
A(1) =2000,
A(2) =1.570296327
A(3) =1.003740589
A(4) =0.787264964
A(5) =0.666927294
A(6) =0.588183016
.
A(40) =0.19864458
.
收敛很慢.