(1)

\[ x_{n}=\sqrt[a_{1}]{r_{1}\sqrt[a_{2}]{r_{2}\cdots\sqrt[a_{n}]{r_{n}}}}\cnd{n\in\mathbb{N}} \] とおくと、
\[ x_{0}=1 \] \[ x_{n}=x_{n-1}r_{n}^{\prod_{k=1}^{n}\frac{1}{a_{k}}} \] より、
\begin{align*} x_{n} & =\exp\Log\left(x_{n}\right)\\ & =\exp\Log\left(x_{n-1}r_{n}^{\prod_{k=1}^{n}\frac{1}{a_{k}}}\right)\\ & =\exp\left\{ \Log\left(x_{n-1}\right)+\Log\left(r_{n}^{\;\prod_{k=1}^{n}\frac{1}{a_{k}}}\right)\right\} \\ & =\exp\left\{ \Log\left(x_{0}\prod_{k=1}^{n-1}\frac{x_{k}}{x_{k-1}}\right)+\Log\left(r_{n}\right)\prod_{k=1}^{n}\frac{1}{a_{k}}\right\} \\ & =\exp\left\{ \sum_{k=1}^{n-1}\Log\left(\frac{x_{k}}{x_{k-1}}\right)+\Log\left(r_{n}\right)\prod_{k=1}^{n}\frac{1}{a_{k}}\right\} \\ & =\exp\left\{ \sum_{k=1}^{n-1}\Log\left(r_{k}^{\;\prod_{j=1}^{k}\frac{1}{a_{j}}}\right)+\Log\left(r_{n}\right)\prod_{k=1}^{n}\frac{1}{a_{k}}\right\} \\ & =\exp\left\{ \sum_{k=1}^{n-1}\left(\Log\left(r_{k}\right)\prod_{j=1}^{k}\frac{1}{a_{j}}\right)+\Log\left(r_{n}\right)\prod_{j=1}^{n}\frac{1}{a_{j}}\right\} \\ & =\exp\left\{ \sum_{k=1}^{n}\left(\Log\left(r_{k}\right)\prod_{j=1}^{k}\frac{1}{a_{j}}\right)\right\} \end{align*}

(2)

\begin{align*} \sqrt[a_{1}]{r\sqrt[a_{2}]{r\cdots\sqrt[a_{n}]{r}}} & =\left[\sqrt[a_{1}]{r_{1}\sqrt[a_{2}]{r_{2}\cdots\sqrt[a_{n}]{r_{n}}}}\right]_{r=r_{1}=\cdots=r_{n}}\\ & =\left[\exp\left\{ \sum_{k=1}^{n}\left(\Log\left(r_{k}\right)\prod_{j=1}^{k}\frac{1}{a_{j}}\right)\right\} \right]_{r=r_{1}=\cdots=r_{n}}\\ & =\exp\left\{ \sum_{k=1}^{n}\left(\Log\left(r\right)\prod_{j=1}^{k}\frac{1}{a_{j}}\right)\right\} \\ & =\pow\left(r,\sum_{k=1}^{n}\prod_{j=1}^{k}\frac{1}{a_{j}}\right) \end{align*}

(3)

\begin{align*} \sqrt[a]{r_{1}\sqrt[a]{r_{2}\cdots\sqrt[a]{r_{1}\sqrt[a]{r_{2}}}}} & =\left[\exp\left\{ \sum_{k=1}^{\infty}\left(\Log\left(r_{k}\right)\prod_{j=1}^{k}\frac{1}{a_{j}}\right)\right\} \right]_{r_{2k-1}=r_{1}\;,\;r_{2k}=r_{2}\;,\;a_{j}=a}\\ & =\exp\left\{ \sum_{k=1}^{\infty}\left(\Log\left(r_{1}\right)\prod_{j=1}^{2k-1}\frac{1}{a}\right)+\sum_{k=1}^{\infty}\left(\Log\left(r_{2}\right)\prod_{j=1}^{2k}\frac{1}{a}\right)\right\} \\ & =\pow\left(r_{1},\sum_{k=1}^{\infty}\frac{1}{a^{2k-1}}\right)\pow\left(r_{2},\sum_{k=1}^{\infty}\frac{1}{a^{2k}}\right)\\ & =\pow\left(r_{1},\frac{a^{-1}}{1-a^{-2}}\right)\pow\left(r_{2},\frac{a^{-2}}{1-a^{-2}}\right)\\ & =\pow\left(r_{1},\frac{a}{a^{2}-1}\right)\pow\left(r_{2},\frac{1}{a^{2}-1}\right) \end{align*}

(4)

\begin{align*} x_{n} & =\sqrt[a]{r\sqrt[a]{r\sqrt[a]{r}\cdots\sqrt[a]{r}}}\\ & =\pow\left(r,\sum_{k=1}^{n}\prod_{j=1}^{k}\frac{1}{a}\right)\\ & =\pow\left(r,\sum_{k=1}^{n}\frac{1}{a^{k}}\right)\\ & =\pow\left(r,\frac{1-\frac{1}{a^{n}}}{a-1}\right) \end{align*}

(5)

(4)より、
\begin{align*} \sqrt[a]{r\sqrt[a]{r\sqrt[a]{r}\cdots}} & =\lim_{n\rightarrow\infty}x_{n}\\ & =\lim_{n\rightarrow\infty}\pow\left(r,\frac{1-\frac{1}{a^{n}}}{a-1}\right)\\ & =\lim_{n\rightarrow\infty}\pow\left(r,\frac{1}{a-1}\right)\\ & =r^{\frac{1}{a-1}} \end{align*}

(5)-2

\[ x=\sqrt[a]{r\sqrt[a]{r\sqrt[a]{r}\cdots}} \] とおくと、
\begin{align*} x & =\sqrt[a]{r\sqrt[a]{r\sqrt[a]{r}\cdots}}\\ & =\sqrt[a]{rx}\\ & =x^{\frac{1}{a}}r^{\frac{1}{a}} \end{align*} \[ x^{\frac{a-1}{a}}=r^{\frac{1}{a}} \] 故に、
\[ x=r^{\frac{1}{a-1}} \]

(6)

\begin{align*} \sqrt[1]{r\sqrt[2]{r\sqrt[3]{r\cdots\sqrt[n]{r}}}} & =\pow\left(r,\sum_{k=1}^{n}\prod_{j=1}^{k}\frac{1}{j}\right)\\ & =\pow\left(r,\sum_{k=1}^{n}\frac{1}{k!}\right)\\ & =\pow\left(r,\frac{e\Gamma\left(n+1,1\right)}{\Gamma\left(n+1\right)}-1\right) \end{align*}

(7)

\begin{align*} \sqrt[1]{r\sqrt[2]{r\sqrt[3]{r\cdots}}} & =\pow\left(r,\sum_{k=1}^{\infty}\prod_{j=1}^{k}\frac{1}{j}\right)\\ & =\pow\left(r,\sum_{k=1}^{\infty}\frac{1}{k!}\right)\\ & =\pow\left(r,e-1\right)\\ & =r^{e-1} \end{align*}