Uniform Convergence: Series Homework Help

[Somefantastik]

Homework Statement

Show uniform convergence

$$\frac{4b}{\pi} \sum^{\inf}_{n=1} \frac{1-(-1)^{n}}{n^{2}}cos(nt)cos(nx)$$

for fixed t

Homework Equations

The Attempt at a Solution

$$\left| cos(nt) \right| \leq 1$$

$$\left| cos(nx) \right| \leq 1$$

$$lim \left|\frac{1 - (-1)^{n}}{n^{2}}\right| \ = \ 0$$

What's next?

 

[Feldoh]

Well it's an alternating series isn't it? So... see if it's conditionally convergent and absolutely convergent

 

[Somefantastik]

I'd like to use the Weierstrauss M test. Can someone help me in that direction?

 

[grief]

What's b? It seems like there is a lot of stuff they added here to try to confuse you. Let's ignore the constant 4b/pi for now. Then for any n, for all x the absolute value of the nth term is less than 2/n^2. We know that the series 2/n^2 converges, so this series also converges.

 

[Dick]

I'd like to use the Weierstrauss M test. Can someone help me in that direction?

Then do so. For each n you want a M_n such the |f_n|<=M_n (where f_n is the nth function in your series) and the series of the M_n converges. It's really pretty straightforward. You've already handled the cos parts. Any suggestions?

 

[wangchong]

This is not an alternative series! The terms with n even are all zeros and terms with n odd are 2/n^2 * cos ( ) * cos (). So take its absolute value and compare it with the series sum(2/n^2, n=1..infty) (I'm ignoring the constant outside the summation. Hence the series is absolutely convergent!

 

[Somefantastik]

Ok thanks everybody! I think I have gleaned enough to get a good answer.