Chapter 5.4, Problem 66ES

Which of the following are valid identities?

(a) ${\displaystyle \underset{i=1}{\overset{n}{\sum }}{a}_i{b}_i}={\displaystyle \underset{i=1}{\overset{n}{\sum }}{a}_i}{\displaystyle \underset{i=1}{\overset{n}{\sum }}{b}_i}$

(b) ${\displaystyle \underset{i=1}{\overset{n}{\sum }}{a}_i^2}={\left({\displaystyle \underset{i=1}{\overset{n}{\sum }}{a}_i}\right)}^2$

(c) ${\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{a_i}{b_i}}=\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{a}_i}}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{b}_i}}$

(d) ${\displaystyle \underset{i=1}{\overset{n}{\sum }}{a}_i}={\displaystyle \underset{j=0}{\overset{n-1}{\sum }}{a}_j{}_{+1}}$