Series converges to a function that satisfies the wave equation

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Problem: show that the series [tex]\sum(1/n^2)*sin(nx)*exp(-ny)[/tex] converges to a continuous function u(x,y),

Then show that U satisfies Uxx + Uyy = 0

Attempt: By the M-test, I know it converges, but I have to find the function it converges to. I tried to simplify the sum by using an identity (euler's), but it didn't make sense because this is not complex. I know the solution should look something like: (1/2)*(f(x+cy)+f(x-cy)), but I don't know how to get there. please help!!!! I've been trying to figure this out for hours!
 

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Thanks!
 

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