How I can solve an infinite system of equations?

Assume nMax=5

```
term(n_)(u_, v_) :=
Sqrt( (Cosh(v) - Cos(u))) A(n) LegendreP(n - 1/2, Cosh(v)) Cos(n u);
partialSum(nMax_)(u_, v_) := Sum(term(n)(u, v), {n, 0, nMax});
```

Differentiating this w.r.t. ν at ν=ν0. Requiring this derivative to be zero. Now how to determine the A(n) coefficients by solving a system of equations and then after replacing A(n) into the equation V(v,u), how we are able to see the V(v,u) result?