OK, with your new formula I'm able to give an incomplete answer now. The difficulty in implementing the forumla
$$-\sum _{n=1}^{\infty } \frac{B_n(1) f^{(n-1)}(0)}{n!}$$
is how to symbolically compute the n-th derivative, which is discussed here. Use the solution in that post, we can easily obtain this:
ramanujanSum[f_] :=
Block[{x, n},
FullSimplify[
-Sum[BernoulliB[n, 1]/n SeriesCoefficient[f[x], {x, 0, n - 1}], {n, ∞}],
n >= 1]]
Notice that f should be a functional relation.
It manages to handle some of the Ramanujan summation mentioned in the corresponding wikipedia page:
ramanujanSum[# &]
(* -1/12 *)
ramanujanSum[1 &]
(* -1/2 *)
Assuming[{k ∈ Integers, k > 0}, ramanujanSum[#^(2 k - 1) &]]
(* -BernoulliB[2 k, 1]/(2 k) *)
Assuming[{k ∈ Integers, k > 0}, ramanujanSum[#^(2 k) &]]
(* 0 *)
But fails in others:
ramanujanSum[1/# &]
(* The output is wrong, which is expected:
the implemented formula doesn't apply to this sequence. *)
trouble = ramanujanSum[(-1)^(# - 1) &]
(* Sadly Sum seems not to be able to handle the final summation. *)
How can one improve it? I've no idea at the moment.
BTW, I doubt if the Ramanujan summation for $(-1)^{n-1}$ is (as stated in the wikipedia page) $1/2$:
trouble /. Sum -> NSum // Quiet
(* 0.5 + 0.31831 I *)
