Proving Continuity of a Fourier Series Function

[quasar987]

I'm puzzled and don't know where to begin with this question; it goes like

"Consider the function f:R²-->R defined by

$$f(x,y) = \sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}sin(nx)sin(ny)$$

Show that f is continuous."

Any hint?

.

 

[Brad Barker]

if you can use the fact that a linear combination of continuous functions is continuous, then the problem is greatly simplified--it becomes a matter of showing that sin(nx)sin(xy) is continuous for any n.


...maybe?

 

[quasar987]

Hey Brad, I appreciate your interest in my problem, however, I don't think an infinite sum qualifies as a linear combination. Nevertheless, your idea made me remember a theorem of 1 variable analysis and I was able to generalize it to an n variable function which solves the problem. :)

 

[Brad Barker]

glad i could be of indirect service!

 

[quasar987]

Next they say "evaluate $f(3\pi / 4, -5\pi /4)$". I realize that this is just

$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} sin^2(\frac{3n \pi }{4})$$

but how do I find the sum? I tried squeezing the sum btw 0 and $\sum (-1)^n/n^2$ but this sum is not 0, so I can't conclude. After this attemp I'm all out of idea. :grumpy:
Any help welcome.

 

[quasar987]

Solved.
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