This note is devoted to a fine study of the convergence of some weighted quadratic and cubic variations of a fractional Brownian motion B with Hurst index H in (0,1/2). With the help of Malliavin calculus, we show that, correctly renormalized, the weighted quadratic variation of B that we consider converges in L^2 to an explicit limit when H<1/4, while we conjecture that it converges in law when H>1/4. In the same spirit, we also show that, correctly renormalized, the weighted cubic variation of B converges in L^2 to an explicit limit when H<1/6.