putnam120,if I change the g to be a uniformly continuous function,can I claim that \sum g(f_n) converges uniformly?
Or it will be the case that convergence/uniform convergence of a series is not invariant under any continuous/uniform continuous function?
Going by what I think you mean you're misunderstanding your question:
If and are such that unif. on and is unif. cont. on then unif. on . Applying this result to the partial sums of a series, say (which converges unif.) we get that uniformly. Thus unif. continuity does preserve unif. convergence. What you're asking however is convergence of the series given by . Do you see the difference?
PS. I think this works for your original question:
Let such that be a unif. conv. series of functions such that it converges absolutely for all , and be unif. continous and for all then converges unif. on and absolutely for all