| | \(\displaystyle \sum _{ k=0 }^{ \infty }{ k{ r }^{ k } } =\left\{ 0+r+2{ r }^{ 2 }+3{ r }^{ 3 }+\cdots +n{ r }^{ n } \right\} \quad \Leftrightarrow \quad \sum _{ k=0 }^{ \infty }{ r\cdot k{ r }^{ k-1 } } =\left\{ 0+r+2{ r }^{ 2 }+3{ r }^{ 3 }+\cdots +n{ r }^{ n } \right\} \) | | \(\displaystyle \sum _{ k=0 }^{ \infty }{ k{ r }^{ k } } =\left\{ 0+r+2{ r }^{ 2 }+3{ r }^{ 3 }+\cdots +n{ r }^{ n } \right\} \quad \Leftrightarrow \quad \sum _{ k=0 }^{ \infty }{ r\cdot k{ r }^{ k-1 } } =\left\{ 0+r+2{ r }^{ 2 }+3{ r }^{ 3 }+\cdots +n{ r }^{ n } \right\} \) |
![[PukiWiki]](http://unitylabo.s601.xrea.com/xoops/modules/xpwiki/image/pukiwiki.png "[PukiWiki]")
