Question

如果 $\sum_{n=0}^{\infty} \cos^{2n} \theta = 5\text{，}$ 那么 $\cos 2\theta$ 的值是多少？

If $\sum_{n=0}^{\infty} \cos^{2n} \theta = 5,$ what is the value of $\cos 2\theta?$

$\dfrac{1}{5}$

$\dfrac{2}{5}$

$\dfrac{\sqrt{5}}{5}$

$\dfrac{3}{5}$

$\dfrac{4}{5}$

该级数是公比为 $\cos^2 \theta$ 的等比级数。

The series is geometric with ratio $\cos^2 \theta$

求和得到 $\dfrac{1}{\sin^2 \theta} = 5$ ，再用 $\cos 2\theta = 1 - 2\sin^2 \theta$ 。

Sum to get $\dfrac{1}{\sin^2 \theta} = 5,$ then use $\cos 2\theta = 1 - 2\sin^2 \theta$

该级数首项为 $1$ ，公比为 $\cos^2 \theta$ ，所以和为 $\dfrac{1}{1 - \cos^2 \theta} = \dfrac{1}{\sin^2 \theta} = 5$ 。

因此 $\sin^2 \theta = \tfrac15$ ，并且 $\begin{aligned} \cos 2\theta &= 1 - 2\sin^2 \theta \\ &= 1 - \tfrac25 = \tfrac35 \end{aligned}\text{。}$

所以正确答案是 D。

The series is geometric with first term $1$ and ratio $\cos^2 \theta,$ so its sum is $\dfrac{1}{1 - \cos^2 \theta} = \dfrac{1}{\sin^2 \theta} = 5.$

Thus $\sin^2 \theta = \tfrac15,$ and $\begin{aligned} \cos 2\theta &= 1 - 2\sin^2 \theta \\ &= 1 - \tfrac25 = \tfrac35. \end{aligned}$

Thus, the correct answer is D.