Calculus 2 Series Question: Prove the inequality

[jrcalcuphysic]

This was already posted by someone else but an answer wasn't received so I thought I'd repost. Any help is appreciated.

Homework Statement

Use the Maclaurin series for cosx and the Alternating Series Estimation Theorem to show that

$$\frac{1}{2} - \frac{x^2}{24} < \frac{1-cosx}{x^2} < \frac{1}{2}$$

Homework Equations

$$cosx = 1 - \frac{x^2}{2} + \frac{x^4}{4} - \cdot \cdot \cdot = \sum_{n=0}^\infty \frac{x^{2n}(-1)^{n}}{(2n)!}$$

The Attempt at a Solution

Using the Alternating Series Estimation Theorem I know the error is less than the next term:

$$|error| < \frac{x^{2n+2}}{(2n + 2)!}$$

I have no idea how to answer this.

 

[micromass]

Can you prove that

$$1-\frac{x^2}{2}<\cos(x)<1-\frac{x^2}{2}+\frac{x^4}{24}$$