Calc 2 series question: prove the inequality

[brushman]

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 then the next term:

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

But I don't know how to use this to prove the inequality.

EDIT: Thanks, I figured it out.

 

[micromass]

Prove first that

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

 

[hunt_mat]

Look at the two functions:
$$f(x):=\frac{1-cosx}{x^2}-\frac{1}{2}\quad g(x)=\frac{1-cosx}{x^2}-\frac{1}{2}+\frac{x^{2}}{24}$$
Look for when the functions f and g have a maximum/ minimum.

 

[micromass]

Look at the two functions:
$$f(x):=\frac{1-cosx}{x^2}-\frac{1}{2}\quad g(x)=\frac{1-cosx}{x^2}-\frac{1}{2}+\frac{x^{2}}{24}$$
Look for when the functions f and g have a maximum/ minimum.

Yes, but that doesn't use MacLaurin series does it?

 

[Whitishcube]

well, you can use the maclaurin series of the function and take the derivative of it as if it was the actual function, no?

 

[hunt_mat]

No it doesn't. All I saw was prove the following inequality :)

 

[jrcalcuphysic]

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 then the next term:

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

But I don't know how to use this to prove the inequality.

EDIT: Thanks, I figured it out.

____________________________________________________________________

I'm having trouble figuring this one out. Brushman, you said you've figured it out - can you help me out?

 

[brushman]

The way I understood it is this:

The first three terms of the series for cosx give us an approximation of the function. We can find this approximation, and the remainder, using the alternating series estimation theorem.

Based on the sign of the remainder, we know whether our approximation was an overestimate, or underestimate of cosx.

If it was an overestimate (the remainder is negative) then we know that our estimation is greater then cosx. For an understimate, we have a positive remainder so are estimation is less then cosx.

This allows us to set up an inequality, the one you see in micromass's post, for which we can then solve for (1-cosx)/(x^2).