Hello!
In the “X-Ray Diffraction” course notes, at the “The reciprocal lattice as a Fourier transform” section, we arrive at the sum \sum_\mathbf{G} = \delta (\mathbf{k} - \mathbf{G}). Following the same derivation in the book done in section 13.1.3, I believe the lecture notes are missing a term of (2 \pi)^3 / V_\text{cell} in front of the sum.