무한급수의 아벨식 변형

$\displaystyle \sum\limits _{{n} = 1} ^{\infty} ( n + 1 ) ^{2} ( a _{n+1} - a _{n} )$ $\displaystyle = \lim\limits _{n \rightarrow \infty} \sum\limits _{k = 1} ^{n} ( k + 1 ) ^{2} ( a _{k+1} - a _{k} )$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\left\{ 2 ^{2} ( a _{2} - a _{1} ) + 3 ^{2} ( a _{3} - a _{2} ) + \cdots + ( n + 1 ) ^{2} ( a _{n+1} - a _{n} ) \right\}}$$\displaystyle = \lim\limits _{n \rightarrow \infty} {\left[ - \left\{ 4 a _{1} + 5 a _{2} + 7 a _{3} + \cdots + ( 2 n + 1 ) a _{n} \right\} + ( n + 1 ) ^{2} a _{n+1} \right]}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\left[ - a _{1} - \left\{ 3 a _{1} + 5 a _{2} + 7 a _{3} + \cdots + ( 2 n + 1 ) a _{n} \right\} \right]}$$\displaystyle + \lim\limits _{n \rightarrow \infty} ( n + 1 ) ^{2} a _{n+1}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\left\{ - a _{1} - \sum\limits _{k = 1} ^{n} ( 2 k + 1 ) a _{k} \right\}}$ ($\displaystyle \because \lim\limits _{n \rightarrow \infty} {n ^{2} a _{n}} = \lim\limits _{n \rightarrow \infty} ( n + 1 ) ^{2} a _{n+1} = 0$) $\displaystyle = - a _{1} - \sum\limits _{n = 1} ^{\infty} {( 2 n + 1 ) a _{n}}$ $\displaystyle = - a _{1} - 2 \sum\limits _{n = 1} ^{\infty} {n a _{n}} - \sum\limits _{n = 1} ^{\infty} a _{n}$ $= - A - 2 B - 1$