Convergence of Sin(x) Series: Understanding the Pattern and Proving Convergence

[cybercrypt13]

Homework Statement

Why does sin(x) - 1/2sin^2(x) + 1/4sin^3(x) - 1/8 sin^4(x) + ... = 2sin(x)/2+sin(x)

How do you know for certain the series converges for all real values of x?

Homework Equations

The Attempt at a Solution

Have no clue where to even start...

Thanks for any help...

 

[dhris]

You should use the standard series identity

$$\sum_{n=0}^{\infty} y^n = \frac{1}{1-y}$$

The series converges for |y|<1. With an appropriate substitution you can make this series appear in your problem.

 

[HallsofIvy]

You might want to consider, separately, what happens when x= $\pi/2$or x= $-\pi/2$.

 

[Gib Z]

To make dhris's hint more obvious, just show that

$$\sum_{n=0}^{\infty} (\frac{- \sin x}{2})^n = \frac{2}{2+ \sin x}$$
then multiply through out by sin x.