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bernoulli is not a "symbolic function" in Sage, so it can't accept inputs that aren't integers like 2*k. We are working on converting many previously only numerical functions in Sage to accept symbolic input (like Bessel functions), but probably the first priority of number theorists in Sage wasn't to allow that particular functionality for Bernoulli numbers. Even if clever power series tricks can allow summations of such things.

So I don't think you can directly do the sum from 1 to infinity this way. However, the magic of lambda functions might make things nearly as good? I don't know what you are looking for exactly.

sage: F = lambda k,n: bernoulli(2*k)/n^(2*k)

But some experimentation suggests that if e.g. the sum for n=1 really does converge to -1/4/(e - e^(3/2)), it must do so VERY slowly, so probably you would only be interested in the symbolic solution, which Maxima doesn't seem to have either.