From "Fourier's series and integrals" by H.S. Carslaw, there is the following question:
Prove the zero locus of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} \sin(n x) \sin(n y) = 0$ is represented by two systems of lines at right angles dividing the $(x,y)$-plane into squares of area $\pi^2$.
Really I have no idea how to prove it. First of all, the $(-1)^{n-1}$ means the sign in front of the sines is changing from positive to negative and back, yes? I don't understand how if $n$ is not changing the sum can be zero? What if all of the terms are positive, or all of the terms are negative? I think so - am I wrong? Also how to prove the claim in question would be interesting. Please help.