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楼主: 王守恩

[原创] 伟大的 "e"

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 楼主| 发表于 2018-6-3 16:12:20 | 显示全部楼层
本帖最后由 王守恩 于 2018-6-3 20:19 编辑


         并不神秘的"e"

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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2018-6-3 19:45:35 | 显示全部楼层
本帖最后由 王守恩 于 2018-6-3 20:17 编辑


                           \(\D “e”\ 连分数展开式中的节点数字说明\)

\(\D e^{\frac{1}{1}}的连分数=1,00,1,1,02,1,1,04,1,1,06,1,1,08,1,1,10,......\)

\(\D e^{\frac{1}{2}}的连分数=1,01,1,1,05,1,1,09,1,1,13,1,1,17,1,1,21,......\)

\(\D e^{\frac{1}{3}}的连分数=1,02,1,1,08,1,1,14,1,1,20,1,1,26,1,1,32,......\)

\(\D e^{\frac{1}{4}}的连分数=1,03,1,1,11,1,1,19,1,1,27,1,1,35,1,1,43,......\)

\(\D e^{\frac{1}{5}}的连分数=1,04,1,1,14,1,1,24,1,1,34,1,1,44,1,1,54,......\)

\(\D e^{\frac{1}{6}}的连分数=1,05,1,1,17,1,1,29,1,1,41,1,1,53,1,1,65,......\)

\(\D e^{\frac{1}{7}}的连分数=1,06,1,1,20,1,1,34,1,1,48,1,1,62,1,1,76,......\)

\(\D e^{\frac{1}{8}}的连分数=1,07,1,1,23,1,1,39,1,1,55,1,1,71,1,1,87,......\)

\(\D e^{\frac{1}{9}}的连分数=1,08,1,1,26,1,1,44,1,1,62,1,1,80,1,1,98,......\)

.........
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2018-6-4 06:58:08 | 显示全部楼层
王守恩 发表于 2018-6-3 19:45
\(\D “e”\ 连分数展开式中的节点数字说明\)

\(\D e^{\frac{1}{1}}的 ...

伟大的 "e"

          \(\D e=2^{{(1 -\frac{1}{2}+\frac{1}{3} -\frac{1}{4}+\frac{1}{5} -\frac{1}{6}+\frac{1}{7} -\frac{1}{8}+\frac{1}{9} -\frac{1}{10}+\frac{1}{11} -\frac{1}{12}......)}^{-1}}\)

          \(\D e=3^{{(1+\frac{1}{2} -\frac{2}{3}+\frac{1}{4}+\frac{1}{5} -\frac{2}{6}+\frac{1}{7}+\frac{1}{8} -\frac{2}{9}+\frac{1}{10}+\frac{1}{11} -\frac{2}{12}......)}^{-1}}\)

          \(\D e=4^{{(1+\frac{1}{2}+\frac{1}{3} -\frac{3}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7} -\frac{3}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11} -\frac{3}{12}......)}^{-1}}\)

          \(\D e=5^{{(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4} -\frac{4}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9} -\frac{4}{10}+\frac{1}{11}+\frac{1}{12}......)}^{-1}}\)

          \(\D e=6^{{(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5} -\frac{5}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11} -\frac{5}{12}......)}^{-1}}\)

          \(\D e=7^{{(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6} -\frac{6}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}......)}^{-1}}\)

          \(\D e=8^{{(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7} -\frac{7}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}......)}^{-1}}\)

          \(\D e=9^{{(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8} -\frac{8}{9}+\frac{1}{10}+\frac{1}{11}+\frac{1}{12}......)}^{-1}}\)

          \(......\)

毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2018-6-5 12:44:30 | 显示全部楼层
本帖最后由 王守恩 于 2018-6-5 14:10 编辑
王守恩 发表于 2018-6-4 06:58
伟大的 "e"

          \(\D e=2^{{(1 -\frac{1}{2}+\frac{1}{3} -\frac{1}{4}+\frac{1}{5} -\frac{1}{ ...




    \(\D n\ 是自然数 ,x\ 是任意数\)

\(\D求证:\sum_{i=0}^n\  (-1)^i\  C_n^i\  (x-i)^n\ =\ n\ !\)


\(\D 1×0^0=0!\)

\(\D 1×1^1-1×0^1=1!\)

\(\D 1×2^2-2×1^2+01×0^2=2!\)

\(\D 1×3^3-3×2^3+03×1^3-01×0^3=3!\)

\(\D 1×4^4-4×3^4+06×2^4-04×1^4+01×0^4=4!\)

\(\D 1×5^5-5×4^5+10×3^5-10×2^5+05×1^5-01×0^5=5!\)

\(\D 1×6^6-6×5^6+15×4^6-20×3^6+15×2^6-06×1^6+01×0^6=6!\)

\(\D 1×7^7-7×6^7+21×5^7-35×4^7+35×3^7-21×2^7+07×1^7-1×0^7=7!\)

\(\D 1×8^8-8×7^8+28×6^8-56×5^8+70×4^8-56×3^8+28×2^8-8×1^8+1×0^8=8!\)



毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2018-6-5 18:27:45 | 显示全部楼层
王守恩 发表于 2018-6-5 12:44
\(\D n\ 是自然数 ,x\ 是任意数\)

\(\D求证:\sum_{i=0}^n\  (-1)^i\  C_n^i\  (x-i)^n\ = ...

         




              \({\large e }\approx\left(\frac{3}{2}\right)^{\sqrt[3]{3×2.5×2}}\approx\left(\frac{4}{3}\right)^{\sqrt[3]{4×3.5×3}}\approx\left(\frac{5}{4}\right)^{\sqrt[3]{5×4.5×4}}\)

                \(\approx\left(\frac{6}{5}\right)^{\sqrt[3]{6×5.5×5}}\approx\left(\frac{7}{6}\right)^{\sqrt[3]{7×6.5×6}}\approx\left(\frac{8}{7}\right)^{\sqrt[3]{8×7.5×7}}\)

             \(\approx\left(\frac{9}{8}\right)^{\sqrt[3]{9×8.5×8}}\approx\left(\frac{10}{9}\right)^{\sqrt[3]{10×9.5×9}}\approx\left(\frac{11}{10}\right)^{\sqrt[3]{11×10.5×10}}\)

               \(......\)

              \(\left(\frac{8}{7}\right)^{\sqrt[3]{8×7.5×7}}=2.71828\)

              \(\left(\frac{46}{45}\right)^{\sqrt[3]{46×45.5×45}}=2.718281828\)






毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2018-6-5 19:17:55 | 显示全部楼层
王守恩 发表于 2018-6-5 18:27
\({\large e }\approx\left(\frac{3}{2}\right)^{\sqrt[3]{3×2.5×2 ...


$$这跟\left(1+\frac{1}{n}\right)^{n}\approx e有什么区别么?$$

点评

精确度快多了。  发表于 2018-6-5 19:22
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2018-6-6 10:45:35 | 显示全部楼层
本帖最后由 王守恩 于 2018-6-6 12:00 编辑
aimisiyou 发表于 2018-6-5 19:17
$$这跟\left(1+\frac{1}{n}\right)^{n}\approx e有什么区别么?$$



       底数不一定是 (n+1)/n,底数可以是任意数!
我是说:千军万马都在向 “e” 靠拢!例如 ”7“。



         \({\large e }\approx\left(\frac{1}{7}\right)^{\sqrt[3]{1×4×7}/(-6)}\approx\left(\frac{2}{7}\right)^{\sqrt[3]{2×4.5×7}/(-5)}\approx\left(\frac{3}{7}\right)^{\sqrt[3]{3×5×7}/(-4)}\)

           \(\approx\left(\frac{4}{7}\right)^{\sqrt[3]{4×5.5×7}/(-3)}\approx\left(\frac{5}{7}\right)^{\sqrt[3]{5×6×7}/(-2)}\approx\left(\frac{6}{7}\right)^{\sqrt[3]{6×6.5×7}/(-1)}\)

              \(\approx\left(\frac{8}{7}\right)^{\sqrt[3]{8×7.5×7}/1}\approx\left(\frac{9}{7}\right)^{\sqrt[3]{9×8×7}/2}\approx\left(\frac{10}{7}\right)^{\sqrt[3]{10×8.5×7}/3}\)

            \(\approx\left(\frac{11}{7}\right)^{\sqrt[3]{11×9×7}/4}\approx\left(\frac{12}{7}\right)^{\sqrt[3]{12×9.5×7}/5}\approx\left(\frac{13}{7}\right)^{\sqrt[3]{13×10×7}/6}\)

             \(.......\)


毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2018-6-6 13:03:38 | 显示全部楼层
aimisiyou 发表于 2018-6-5 19:17
$$这跟\left(1+\frac{1}{n}\right)^{n}\approx e有什么区别么?$$

还是“7”!效率就高了!!


         \({\large e }\approx\left(\frac{6.2}{7}\right)^{\sqrt[3]{6.2×6.6×7}/(-0.8)}\approx\left(\frac{6.5}{7}\right)^{\sqrt[3]{6.5×6.75×7}/(-0.5)}\approx\left(\frac{6.8}{7}\right)^{\sqrt[3]{6.8×6.9×7}/(-0.2)}\)


         \(\approx\left(\frac{6.9}{7}\right)^{\sqrt[3]{6.9×6.95×7}/(-0.1)}\approx\left(\frac{6.99}{7}\right)^{\sqrt[3]{6.99×6.995×7}/(-0.01)}\approx\left(\frac{6.999}{7}\right)^{\sqrt[3]{6.999×6.9995×7}/(-0.001)}\)


         \(\approx\left(\frac{7.001}{7}\right)^{\sqrt[3]{7.001×7.0005×7}/0.001}\approx\left(\frac{7.01}{7}\right)^{\sqrt[3]{7.01×7.005×7}/0.01}\approx\left(\frac{7.1}{7}\right)^{\sqrt[3]{7.1×7.05×7}/0.1}\)


         \(\approx\left(\frac{7.2}{7}\right)^{\sqrt[3]{7.2×7.1×7}/0.2}\approx\left(\frac{7.5}{7}\right)^{\sqrt[3]{7.5×7.25×7}/0.5}\approx\left(\frac{7.8}{7}\right)^{\sqrt[3]{7.8×7.4×7}/0.8}\)

             \(.......\)





毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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