Exercise 2.8.7

Assume that ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{\infty }{a}_i$ converges absolutely to $A$, and ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{j=1}^{\infty }{b}_j$ converges absolutely to $B$.

(a)

Show that the iterated sum ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{\infty }{\sum }_{j=1}^{\infty }|a_i{b}_j|$ converges so that we may apply Theorem 2.8.1.

(b)

Let $s_{\mathit{nn}}={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^n{\sum }_{j=1}^n{a}_i{b}_j$, and prove that $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{s}_{\mathit{nn}}=\mathit{AB}$. Conclude that

$$\underset{i=1}{\overset{\infty }{\sum }}\underset{j=1}{\overset{\infty }{\sum }}{a}_i{b}_j=\underset{j=1}{\overset{\infty }{\sum }}\underset{i=1}{\overset{\infty }{\sum }}{a}_i{b}_j=\underset{k=2}{\overset{\infty }{\sum }}{d}_k=\mathit{AB},$$

where, as before, $d_k=a_1{b}_{k-1}+a_2{b}_{k-2}+\cdots +a_{k-1}{b}_1$.