09/10/17 22:14:51
>>941
a[n+1]↑ は a[1]↑ /(2^n) を原点の周りに角nθ回転したもの。
a[n+1]↑ = ((1/2^n)cos(nθ), (1/2^n)sin(nθ)),
x_n +i･y_n = Σ[k=0,n-1] (1/(2^k)){cos(kθ)+i･sin(kθ)}
= Σ[k=0,n-1] {(1/2)exp(iθ)}^k (← 等比級数)
= {1 - [(1/2)exp(iθ)]^n} / {1-(1/2)exp(iθ)}
→ 1/{1-(1/2)exp(iθ)}
= {1-(1/2)exp(-iθ)}/{(5/4)-cosθ}, (n→∞)
実部と虚部に分けて
x_n → {1-(1/2)cosθ}/{(5/4)-cosθ}, (n→∞)
y_n → (1/2)sinθ/{(5/4)-cosθ}, (n→∞)
∴ 点Pは円 (x - 4/3)^2 + y^2 = (2/3)^2 の周上を動く。