题目

数列极限
a1=(1/2)^(1/2),an=[(1+ an-1)/2]^(1/2),求lim (a1a2a3.an) (n趋于正无穷大)

提问时间：2021-01-29

答案

考虑余弦半角公式：cos(θ/2)=√((1+cosθ)/2)令a[n]=cos(θ/2^n),则a[1]=cos(θ/2)=√2/2所以θ=π/2a[n]=cos(π/2^(n+1))a[1]*a[2]*..*a[n]=cos(π/4)*cos(π/8)*...*cos(π/2^(n+1))=2^n*sin(π/2^(n+1))*cos(π/...