notation for this summation I have a summation of the product of two variable like this $v_{j}f_i$ where
$j=1,2,\dots,d,d+1,\dots, 2d,\dots, nd$ and after each $d$ step $i$ changes from $1$ to $n$, is my notation following correct in this situation?
$$\sum_{j=1}^{nd}\sum_{i=1}^{n} v_j f^{(i)}?$$
what I mean is I want $(v_1+\dots+v_d)f^{(1)}+(v_{d+1}+\dots+v_{2d})f^{(2)}+\dots$

So you have
$$
\begin{split}
S &= (v_1+\dots+v_d)f^{(1)}+(v_{d+1}+\dots+v_{2d})f^{(2)}+\dots \\
  &= f^{(1)}\sum_{k=1}^d v_k + f^{(2)}\sum_{k=d+1}^{2d} v_k + \ldots \\
  &= \sum_{i=0}^n f^{(i+1)}\sum_{k=di+1}^{d(i+1)} v_k
\end{split}
$$
