I just don't understand what is the relationship between example (1) and this problem

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This is my pde homework enter image description here

whose hint is the example (1) enter image description here

after reading the whole chapter, I still do not know how this example has anything to do with my homework

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The example seems to be about Fourier coefficients. Under some conditions (you should check that they are satisfied when you want to apply this!) the Fourier series is equal to the original function. Maybe you can find a function $f$ so that its Fourier series at a point $x$ has the form $\sum_{n\text{ odd}}C\frac1{n^2}$ for some constant $C$. Then you could calculate the value of this series via $f(x)$ if the function $f$ is equal to its Fourier series.