方程 $\cot\frac12x=3x$ 在区间 $0<x<\pi$ 内有一个根，记为 $\alpha$。

(a) 通过计算证明 $\alpha$ 位于 $0.5$ 与 $1$ 之间。

(b) 证明：若由迭代公式

给出的正数数列收敛，则它收敛于 $\alpha$。

(c) 使用此迭代公式求 $\alpha$，结果保留 $2$ 位小数。每次迭代的结果保留 $4$ 位小数。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

构造两边之差，在指定两点计算异号值。注意所有角均用弧度。

Let $f(x)=\cot(x/2)-3x$. Then, in radians,

Since $f$ is continuous on $[0.5,1]$, it has a root there. This is the given unique root $\alpha$, so $\boxed{0.5<\alpha<1}$.

设极限为 $L$。先排除正数列趋于零的可能，再在迭代式中取极限，整理并取正切，得到原方程。

Suppose the positive sequence converges to $L\ge0$. If $L=0$, then

so the recurrence would give $x_{n+1}\to2\pi/3$, a contradiction. Thus $L>0$.

Taking limits,

Hence

Since $L>0$, the right side is in $(0,\pi/2)$, so $0<L<\pi$. Taking tangents,

Thus $L$ is the root in $(0,\pi)$ and therefore $\boxed{L=\alpha}$.

按给定迭代式计算，保留内部精度，表中写到四位小数；继续到稳定值以判断两位小数。

Starting with $x_0=1$, use the stated formula in radians.

The values converge to a number which rounds to $0.79$. Hence