Did I evaluate the following terms correctly? Does the set notation in example b) allow me to chose the order of the terms?
$$ a) \sum_{i=1}^6 ix^{i+1} = x^2+2x^3+3x^4+4x^5+5x^6+6x^7 \\ b) \prod_{i \in \{5, 6, 7\}} (ix-3) =(5x-3)(6x-3)(7x-3) \\ c) \sum_{j=3}^6 \prod_{k=1}^3 (jk-2) = \sum_{j=3}^6 (j1-2)(j2-2)(j3-2) = (3*1-2)(3*2-2)(3*3-2)+(4*1-2)(4*2-2)(4*3-2)+...+(6*1-2)(6*2-2)(6*3-2) \\ d) \sum_{k=0}^m \sum_{j=k}^m k^j = \sum_{k=0}^m k^k+k^{k+1}+...+k^{m-1}+k^m= 0^0+0^1+...+0^{m-1}+0^m+1^0+1^1+...+1^{m-1}+1^m+...+m^k+m^{k+1}+...+m^{m-1}+m^m $$
a), b) and c) look fine for me. But I don't agree with you for d). If you review it, you're going to see that in fact:
$$\sum_{k=0}^m \sum_{j=k}^m k^j = 0^0+0^1+...+0^{m-1}+0^m+1^1+...+1^{m-1}+1^m+\cdots+m^m $$