Chapter 8.2, Problem 28E

Show that

$\begin{array}{rllll}2{\displaystyle \underset{n}{\overset{}{\sum }}}& =& 0\infty a_n{x}^{n+1}+{\displaystyle \underset{n}{\overset{}{\sum }}}& =& 1\infty n{b}_n{x}^{n-1}\\ & =& b_1+{\displaystyle \underset{n}{\overset{}{\sum }}}& =& 1\infty \left[2a_{n-1}+\left(n+1\right)b_{n+1}\right]x^n\end{array}$.