Exercise
14
[217]Suppose \((a_ n)_ n,(b_ n)_ n\) are sequences of real numbers and \(c_ n\) is defined by [0FH]; let then
\[ A_ n=β_{h=0}^ n a_ h~ ~ ,~ ~ B_ n =β_{h=0}^ n b_ h ~ ~ ,~ ~ C_ n=β_{h=0}^ n c_ h \]
the partial sums of the three series; suppose that \(β_{n=0}^β b_ n=B\) is convergent: show that
\[ C_ n=β_{i=0}^ n a_{n-i}B_ i=β_{i=0}^ n a_{n-i}(B_ i-B)+A_ nB \quad . \]
Solution
1