Show that $\sum_{n=1}^\infty 2^{2n}\sin^4\frac a{2^n}=a^2-\sin^2a$

Question

Show that $$\sum_{n=1}^\infty 2^{2n}\sin^4\frac a{2^n}=a^2-\sin^2a$$

I am studying for an exam and I bumped into this question. It's really bothering me because I really don't have any clue what to do. Does it have anything to do with the Cauchy condensation? Can somebody help me I would really appreciate it

Answer 1

Hint: Use the fact that $\sin^4x=(\sin^2 x)(\sin^2 x)$ and a couple of trigonometric identities to reduce it to a telescopic series. Finally, use the identity $\lim_{x\to0}\frac{\sin x}{x}=1$ to obtain the desired value.

Answer 2

Using the hint that @Clayton gave this is what you get if you take the partial sum

$$2^2\sin^2\frac a{2}-\sin^2a+2^4\sin^2\frac a{2^2}-2^2\sin^2\frac a{2}+ \cdots+ 2^{2n}\sin^2\frac a{2^n}-2^{2n-2}\sin^2\frac a{2^{n-1}}= 2^{2n}\sin^2\frac a{2^n}-\sin^2a$$

when you take the limit $$\lim_{n\to \infty}\left(2^{2n}\sin^2\frac a{2^n}-\sin^2a\right)=a^2-\sin^2a$$

Answer 3

Solution

Notice that \begin{align*} 2^{2n}\sin^4\frac a{2^n}&=2^{2n}\cdot\sin^2\frac a{2^n}\cdot\sin^2\frac a{2^n}\\&=2^{2n}\cdot\sin^2\frac a{2^n}\cdot\left(1-\cos^2\frac a{2^n}\right)\\&=2^{2n}\cdot\sin^2\frac a{2^n}-2^{2n}\cdot\sin^2\frac a{2^n}\cos^2\frac a{2^n}\\&=2^{2n}\cdot\sin^2\frac a{2^n}-2^{2n-2}\sin^2\frac a{2^{n-1}}. \end{align*}

Hence, the partial sum \begin{align*} &\sum_{n=1}^{m}\left(2^{2n}\sin^4\frac a{2^n}\right)\\=&2^2\sin^2\frac a{2}-\sin^2a+2^4\sin^2\frac a{2^2}-2^2\sin^2\frac a{2}+ \cdots+ 2^{2m}\sin^2\frac a{2^m}-2^{2m-2}\sin^2\frac a{2^{m-1}}\\=&2^{2m}\sin^2\frac a{2^m}-\sin^2a. \end{align*}

Let $m \to \infty$. We obtain $$\sum_{n=1}^{\infty}\left(2^{2n}\sin^4\frac a{2^n}\right)=\lim_{m \to \infty}\left(\frac{\sin\dfrac a{2^m}}{\dfrac{a}{2^m}}\cdot a\right)^2-\sin^2 a=(1\cdot a)^2-\sin^2 a=a^2-\sin^2 a.$$