tag:
цепные_дроби
§
[Рамануджан]
\[1 + \frac{1}{{1 \cdot 3}} + \frac{1}{{1 \cdot 3 \cdot 5}} + \frac{1}{{1 \cdot 3 \cdot 5 \cdot 7}} + \frac{1}{{1 \cdot 3 \cdot 5 \cdot 7 \cdot 9}} + \cdot \cdot \cdot + \frac{1}{{1 + \frac{1}{{1 + \frac{2}{{1 + \frac{3}{{1 + \frac{4}{{1 + ...}}}}}}}}}} = \sqrt {\frac{{\pi \cdot e}}{2}} .\]
Примечание
\[\begin{array}{l} S = \sum\limits_{k = 1}^{ + \infty } {\frac{1}{{\left( {2k - 1} \right)!!}}} = \sqrt {\frac{{\pi \cdot e}}{2}} \cdot \left( {1 - \operatorname{erfc} \left( {\frac{{\sqrt 2 }}{2}} \right)} \right) \hfill \\ F = \frac{1}{{1 + \frac{1}{{1 + \frac{2}{{1 + \frac{3}{{1 + ...}}}}}}}} = \sqrt {\frac{{\pi \cdot e}}{2}} \cdot \operatorname{erfc} \left( {\frac{{\sqrt 2 }}{2}} \right) \hfill \\ \operatorname{erfc} x = 1 - \operatorname{erf} x \hfill \\ \operatorname{erf} x = \frac{2}{{\sqrt \pi }}\int\limits_0^x {{e^{ - {t^2}}}dt} \hfill \\ \end{array}\]
1722.
\[\tanh \frac{1}{n} = \frac{1}{{n + \frac{1}{{3n + \frac{1}{{5n + \frac{1}{{7n + \frac{1}{{...}}}}}}}}}}\]
1935.
\[\frac{{{\pi ^2}}}{6} = 1 + \frac{1}{{1 + \frac{1}{{1 + \frac{1}{{\frac{1}{2} + \frac{1}{{\frac{1}{2} + \frac{1}{{\frac{1}{3} + \frac{1}{{\frac{1}{3} + ...}}}}}}}}}}}} = \left[ {1;1,1,\frac{1}{2},\frac{1}{2},\frac{1}{3},\frac{1}{3},\frac{1}{4},\frac{1}{4},\frac{1}{5},\frac{1}{5}...} \right]\]
1949.
\[\frac{1}{{2 + \frac{1}{{\frac{2}{2} + \frac{1}{{\frac{2}{3} + \frac{1}{{\frac{2}{4} + \frac{1}{{\frac{2}{5} + \frac{1}{{...}}}}}}}}}}}} = 2\ln 2 - 1\]
1950.
\[\frac{1}{{3 + \frac{1}{{\frac{3}{2} + \frac{1}{{\frac{3}{3} + \frac{1}{{\frac{3}{4} + \frac{1}{{\frac{3}{5} + \frac{1}{{...}}}}}}}}}}}} = 5 - \frac{{3\pi }}{2}\]
1951.
\[\frac{1}{{4 + \frac{1}{{\frac{4}{2} + \frac{1}{{\frac{4}{3} + \frac{1}{{\frac{4}{4} + \frac{1}{{\frac{4}{5} + \frac{1}{{...}}}}}}}}}}}} = 3 - 4\ln 2\]
1952.
\[1 = \frac{2}{{1 + \frac{2}{{1 + \frac{2}{{1 + \frac{2}{{1 + ...}}}}}}}}\]
1953.
\[e = 2 + \frac{1}{{1 + \frac{1}{{2 + \frac{2}{{3 + \frac{3}{{4 + \frac{4}{{5 + \frac{5}{{...}}}}}}}}}}}}\]
1954.
\[\frac{{{\pi ^2}}}{4} = 2 + \frac{1}{{1 + \frac{1}{2} + \frac{1}{{\frac{1}{2} + \frac{1}{3} + \frac{1}{{\frac{1}{3} + \frac{1}{4} + \frac{1}{{\frac{1}{4} + \frac{1}{5} + \frac{1}{{...}}}}}}}}}}\]
1955.
\[\frac{{{\pi ^2}}}{3} = 3 + \frac{1}{{\frac{2}{1} + \frac{2}{2} + \frac{1}{{\frac{2}{2} + \frac{2}{3} + \frac{1}{{\frac{2}{3} + \frac{2}{4} + \frac{1}{{\frac{2}{4} + \frac{2}{5} + \frac{1}{{...}}}}}}}}}}\]
1962.
\[\frac{{2{\pi ^2}}}{3} = 7 - \frac{1}{{2 + \frac{1}{{2 + \frac{1}{{\frac{2}{2} + \frac{1}{{\frac{2}{2} + \frac{1}{{\frac{2}{3} + \frac{1}{{\frac{2}{3} + ...}}}}}}}}}}}} = 7 - \left[ {0;2,2,\frac{2}{2},\frac{2}{2},\frac{2}{3},\frac{2}{3},\frac{2}{4},\frac{2}{4},\frac{2}{5},\frac{2}{5}...} \right].\]
1963.
\[\begin{array}{l}
\frac{e}{{e - 1}} = 1 + \frac{1}{{1 + \frac{1}{{1 + \frac{1}{{1 + \frac{2}{{1 + \frac{1}{{1 + \frac{3}{{1 + ...}}}}}}}}}}}} \hfill \\
{b_i} = \left[ {1,1,1,2,1,3,1,4,1,5,1,...} \right] \hfill \\
\end{array}\]
1964.
\[\begin{array}{l}
\frac{e}{2} = 1 + \frac{1}{{2 + \frac{1}{{1 + \frac{1}{{2 + \frac{2}{{1 + \frac{1}{{2 + \frac{3}{{1 + ...}}}}}}}}}}}} \hfill \\
{b_i} = \left[ {1,1,1,2,1,3,1,4,1,5,1,...} \right] \hfill \\
\end{array}\]
1965.
\[\begin{array}{l}
\ln 2 = \frac{1}{{1 + \frac{1}{{1 + \frac{2}{{1 + \frac{2}{{1 + \frac{4}{{1 + \frac{3}{{1 + \frac{6}{{...}}}}}}}}}}}}}} \hfill \\
{b_i} = \left[ {1,2,2,4,3,6,4,8,...,n,2n,...} \right] \hfill \\
\end{array}\]
1966.
\[e = 2 + \frac{1}{{1 + \frac{{1/2}}{{1 + \frac{{1/3}}{{1 + \frac{{1/4}}{{1 + \frac{{1/5}}{{1 + ...}}}}}}}}}}\]
1978.
\[\coth x = \frac{{{e^{2x}} + 1}}{{{e^{2x}} - 1}} = \frac{1}{x} + \frac{{\frac{1}{{1 \cdot 3}}}}{{\frac{1}{x} + \frac{{\frac{1}{{3 \cdot 5}}}}{{\frac{1}{x} + \frac{{\frac{1}{{5 \cdot 7}}}}{{\frac{1}{x} + ...}}}}}}.\]
1979.
\[\frac{{{\pi ^2}}}{6} = \frac{1}{1} + \frac{{\frac{1}{2} + \frac{{\frac{1}{3} + \frac{{\frac{1}{4} + \frac{{\frac{1}{5} + ...}}{{\frac{5}{4}}}}}{{\frac{4}{3}}}}}{{\frac{3}{2}}}}}{{\frac{2}{1}}}.\]
1980.
\[\begin{array}{l}
{\text{Пусть }}{a_k} = \frac{{\left( {2k + 1} \right)\left( {2k + 2} \right)}}{{{k^2}}}. \hfill \\
\frac{{{\pi ^2}}}{9} = 1 + \frac{{1 + \frac{{1 + \frac{{1 + ...}}{{{a_3}}}}}{{{a_2}}}}}{{{a_1}}}. \hfill \\
\end{array}\]
1982.
\[\begin{array}{l}
N = \frac{{\sqrt 2 {\pi ^{\frac{3}{2}}}}}{{4\Gamma {{\left( {\frac{3}{4}} \right)}^2}}} = \frac{{\pi \cdot G}}{2} = \frac{1}{4}{\rm B}\left( {\frac{1}{4},\frac{1}{2}} \right). \hfill \\
\Gamma {\text{ - гамма - функция;}} \hfill \\
G{\text{ - постоянная Гаусса;}} \hfill \\
{\rm B}{\text{ - бета - функция}}{\text{.}} \hfill \\
{\text{Докажите}}{\text{, что }}1 + \frac{{1 + \frac{{1 + \frac{{1 + \frac{{1 + ...}}{{2 + 3/7}}}}{{2 + 3/5}}}}{{2 + 3/3}}}}{{2 + 3/1}} = N. \hfill \\
{a_k} = 2 + \frac{3}{{2k - 1}}. \hfill \\
\end{array}\]
1985.
\[\frac{\pi }{2} = 1/1 + \frac{{1/3 + \frac{{1/5 + \frac{{1/7 + ...}}{{6/5}}}}{{4/3}}}}{{2/1}}\]
2068.
\[\begin{array}{l}
{\text{Докажите}}{\text{, что}} \hfill \\
{\left( {\frac{a}{{a - 1}}} \right)^{\frac{{b - 1}}{b}}} = 1 + \frac{{1 + \frac{{1 + \frac{{...}}{{{a_3}}}}}{{{a_2}}}}}{{{a_1}}}, \hfill \\
{\text{где }}{a_k} = a + \frac{a}{{b \cdot k - 1}},{\text{ }}a,b > 1. \hfill \\
\end{array} \]
2072.
$%F = 1 + \frac{{1 + \frac{{1 + \frac{{...}}{{{a_3}}}}}{{{a_2}}}}}{{{a_1}}},{\text{ }}{a_k} = 2 + \frac{4}{{2k - 1}}.$%
2073.
\[\begin{array}{l}
1 + \frac{{1 + \frac{{1 + \frac{{...}}{{{a_3}}}}}{{{a_2}}}}}{{{a_1}}} = 1 + \frac{{\Gamma {{\left( {\frac{1}{4}} \right)}^2}}}{{4\sqrt {2\pi } }}, \hfill \\
{a_n} = 1 + \frac{{2n}}{{2n + 1}}. \hfill \\
\end{array} \]
2136.
$$\begin{array}{l}
\frac{1}{2} + \frac{2}{{1 + 2 \cdot \left( {\frac{1}{4} + \frac{4}{{1 + 4 \cdot \left( {\frac{1}{6} + \frac{6}{{1 + 6 \cdot \left( {...} \right)}}} \right)}}} \right)}} = \frac{{{\pi ^2}}}{8} - 1 + G, \hfill \\
{\text{где }}G{\text{ - постоянная Каталана}}{\text{.}} \hfill \\
\end{array} $$
2138.
$$\frac{{{\pi ^2}}}{6} = \frac{3}{2} + \frac{1}{{7 - \frac{{{1^4}}}{{11 - \frac{{{2^4}}}{{19 - \frac{{...}}{{... - \frac{{{n^4}}}{{2{n^2} - 2n + 7 - ...}}}}}}}}}}$$
2142.
$$\sum\limits_{k = 0}^{ + \infty } {\dfrac{{{x^{k + 1}}}}{{k{!^s}}}} = x + \dfrac{{{x^2} + \dfrac{{{x^3} + \dfrac{{{x^4} + \dfrac{{{x^5} + ...}}{{{4^s}}}}}{{{3^s}}}}}{{{2^s}}}}}{{{1^s}}},{\text{ }}s > 0$$
2145.
$$1 + \cfrac{1}{{1 + \cfrac{2}{{1 + \cfrac{3}{{1 + \cfrac{4}{{1 + ...}}}}}}}} = \cfrac{2}{{1 + \cfrac{1}{{1 + \cfrac{4}{{1 + \cfrac{3}{{... + \cfrac{{2n}}{{1 + \cfrac{{2n - 1}}{{1 + ...}}}}}}}}}}}}$$
2153.
$$1 + \cfrac{{\cfrac{1}{{1 \cdot s}}}}{{1 + \cfrac{{\cfrac{1}{{2 \cdot s}}}}{{1 + \cfrac{{\cfrac{1}{{3 \cdot s}}}}{{1 + ...}}}}}} = 1 + \cfrac{{1 + \cfrac{{1 + \cfrac{{1 + ...}}{{3s + 1}}}}{{2s + 1}}}}{{1s + 1}} = \sum\limits_{n = 0}^{ + \infty } {\frac{1}{{\prod\limits_{k = 1}^n {\left( {sk + 1} \right)} }}} = {s^{\frac{1}{s} - 1}}{e^{\frac{1}{s}}}\left( {\Gamma \left( {\frac{1}{s}} \right) - \Gamma \left( {\frac{1}{s},\frac{1}{s}} \right)} \right)$$
2154.
$$\eqalign{
& s > 0,{\text{ }}s \ne \frac{1}{n}; \cr
& \cfrac{1}{{\cfrac{s}{{1 + \cfrac{{s - 1}}{{1 + \cfrac{{2s}}{{1 + \cfrac{{2s - 1}}{{1 + \cfrac{{3s}}{{1 + \cfrac{{3s - 1}}{{1 + ...}}}}}}}}}}}}}} = {s^{\frac{1}{s} - 1}} \cdot {e^{\frac{1}{s}}} \cdot \Gamma \left( {\frac{1}{s},\frac{1}{s}} \right) \cr} $$
2155.
$$1 + \cfrac{{1 + \cfrac{{1 + \cfrac{{1 + \cfrac{{1 + ...}}{{4s + 1}}}}{{3s + 1}}}}{{2s + 1}}}}{{s + 1}} + \cfrac{1}{{\cfrac{s}{{1 + \cfrac{{s - 1}}{{1 + \cfrac{{2s}}{{1 + \cfrac{{2s - 1}}{{1 + \cfrac{{3s}}{{1 + \cfrac{{3s - 1}}{{1 + ...}}}}}}}}}}}}}} = {s^{\frac{1}{s} - 1}}{e^{\frac{1}{s}}}\Gamma \left( {\frac{1}{s}} \right)$$
\[s > 0,{\text{ }}s \ne \frac{1}{n};\]
2156.
$$\eqalign{
& \cfrac{1}{{1 + x}} + \cfrac{{\cfrac{1}{{1 + x/2}} + \cfrac{{\cfrac{1}{{1 + x/3}} + \cfrac{{...}}{{3/x}}}}{{2/x}}}}{{1/x}} = \cr
& = x \cdot {\left( { - x} \right)^{ - x}} \cdot \left( {\Gamma \left( x \right) - \Gamma \left( {x, - x} \right)} \right) \cr
& = x \cdot {\left( { - x} \right)^{ - x}}\int\limits_0^{ - x} {{t^{x - 1}}{e^{ - t}}dt} = x{\left( { - x} \right)^{ - x}}\gamma \left( {x, - x} \right) \cr} $$