拉马努金恒等式通式（3生万数）

朱明君

\(拉马努金恒等式通式（3生万数）\)

\(3=\sqrt{1+\left( n_1+1\right)\sqrt{1+\left( n_2+1\right)\cdots\sqrt{1+\left( n_n+1\right)\left( n_n+3\right)}}}\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ 其中n_1=1{,}\ \ n_2=2{,}\ \ \cdots\ \ n_n=n_{ }\)

\(3=\sqrt{1+\left( n_1+1\right)\left( n_1+3\right)}=\sqrt{1+2\times4}\)
\(3=\sqrt{1+\left( n_1+1\right)\sqrt{1+\left( n_2+1\right)\left( n_2+3\right)}}=\sqrt{1+2\sqrt{1+3\times5}}\)
\(3=\sqrt{1+\left( n_1+1\right)\sqrt{1+\left( n_2+1\right)\sqrt{1+\left( n_3+1\right)\left( n_3+3\right)=}}}\sqrt{1+2\sqrt{1+3\sqrt{1+4\times6}}}\)
\(3=\sqrt{1+\left( n_1+1\right)\sqrt{1+\left( n_2+1\right)\sqrt{1+\left( n_3+1\right)\sqrt{1+\left( n_4+1\right)\left( n_4+3\right)}}}}=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\times7}}}}\)
\(\cdots\cdots。\)

朱明君

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