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楼主: 王守恩

[灌水] 没有数字的得数

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 楼主| 发表于 2019-1-19 08:04:19 | 显示全部楼层
本帖最后由 王守恩 于 2019-1-19 16:21 编辑
王守恩 发表于 2019-1-17 16:21
对 9 楼的公式作简化。

我们的目标很明确:



左边用电脑也不好算,试试用10楼的算式。

     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{1}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+0)!}=\frac{1441729e}{40320}\)
     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{n_{1}}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+1)!}=\frac{4596553e}{362880}\)
     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{n_{1}\cdot n_{2}}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+2)!}=\frac{1441729e}{403200}\)
     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{n_{1}\cdot n_{2}\cdot n_{3}}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+3)!}=\frac{32989969e}{39916800}\)
     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{n_{1}\cdot n_{2}\cdot n_{3}\cdot n_{4}}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+4)!}=\frac{76751233e}{479001600}\)
     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{n_{1}\cdot n_{2}\cdot n_{3}\cdot n_{4}\cdot n_{5}}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+5)!}=\frac{2628983e}{98841600}\)
     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{n_{1}\cdot n_{2}\cdot n_{3}\cdot n_{4}\cdot n_{5}\cdot n_{6}}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+6)!}=\frac{335262313e}{87178291200}\)
     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{n_{1}\cdot n_{2}\cdot n_{3}\cdot n_{4}\cdot n_{5}\cdot n_{6}\cdot n_{7}}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+7)!}=\frac{642451441e}{1307674368000}\)
     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{n_{1}\cdot n_{2}\cdot n_{3}\cdot n_{4}\cdot n_{5}\cdot n_{6}\cdot n_{7}\cdot n_{8}}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+8)!}=\frac{130469561e}{2324754432000}\)
     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\sum_{n_{9}=0}^{\infty}\frac{n_{1}\cdot n_{2}\cdot n_{3}\cdot n_{4}\cdot n_{5}\cdot n_{6}\cdot n_{7}\cdot n_{8}\cdot n_{9}}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8}+n_{9})!}=e\sum_{m=0}^{8}\frac{8!}{m!(8-m)!(m+9)!}=\frac{158433901e}{27360571392000}\)





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 2019-1-20 17:36:29 | 显示全部楼层
本帖最后由 王守恩 于 2019-1-20 18:26 编辑
王守恩 发表于 2019-1-19 08:04
左边用电脑也不好算,试试用10楼的算式。

     \(\D\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\inf ...


    求助:这有限怎么就替代了无限,能点拨一下吗?

A,B=0, 1, 2, 3, 4, 5,......


\[\D\frac{\D\sum_{m=0}^{A}\D \frac{A!}{m!\ (A-m)!\ (m+B)!}}{\D\sum_{n=0}^{\infty}\ \ \frac{(n+A)!}{n!\ (A+B)!\ \ (n-B)!}}=\D\frac{1}{\D\ e\ }\]

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 2019-1-29 19:48:39 | 显示全部楼层
本帖最后由 王守恩 于 2019-1-29 19:53 编辑
mathematica 发表于 2018-12-30 14:13
这样的问题不要发论坛上了,没人会证的,
真的太难了,又是拉马努金一样的等式,
看到就头疼



右边用电脑也不好算,试试用左边的算式。

\(\D\sum_{m=0}^{0}\frac{0!× e}{m!\ (0-m)!\ (0-m)!\ }=\sum_{n_{1}=0}^{\infty}\frac{1}{n_{1}!\ }\)
\(\D\sum_{m=0}^{1}\frac{1!× e}{m!\ (1-m)!\ (1-m)!\ }=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\frac{1}{(n_{1}+n_{2})!\ }\)
\(\D\sum_{m=0}^{2}\frac{2!× e}{m!\ (2-m)!\ (2-m)!\ }=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\frac{1}{(n_{1}+n_{2}+n_{3})!\ }\)
\(\D\sum_{m=0}^{3}\frac{3!× e}{m!\ (3-m)!\ (3-m)!\ }=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\frac{1}{(n_{1}+n_{2}+n_{3}+n_{4})!\ }\)
\(\D\sum_{m=0}^{4}\frac{4!× e}{m!\ (4-m)!\ (4-m)!\ }=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\frac{1}{(n_{1}+n_{2}+n_{3}+n_{4}++n_{5})!\ }\)
\(\D\sum_{m=0}^{5}\frac{5!× e}{m!\ (5-m)!\ (5-m)!\ }=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\frac{1}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6})!\ }\)
\(\D\sum_{m=0}^{6}\frac{6!× e}{m!\ (6-m)!\ (6-m)!\ }=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\frac{1}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7})!\ }\)
\(\D\sum_{m=0}^{7}\frac{7!× e}{m!\ (7-m)!\ (7-m)!\ }=\sum_{n_{1}=0}^{\infty}\sum_{n_{2}=0}^{\infty}\sum_{n_{3}=0}^{\infty}\sum_{n_{4}=0}^{\infty}\sum_{n_{5}=0}^{\infty}\sum_{n_{6}=0}^{\infty}\sum_{n_{7}=0}^{\infty}\sum_{n_{8}=0}^{\infty}\frac{1}{(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}+n_{7}+n_{8})!\ }\)



   
   
   
   
   

   



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 2019-1-30 14:41:09 | 显示全部楼层
本帖最后由 王守恩 于 2019-1-30 16:12 编辑

右边很好算,左边用电脑也不好算。
2, 5, 10, 17, 26, 37, 50, 65, 82, 101, 122, 145, 170, 197, 226,
257, 290, 325, 362, 401, 442, 485, 530, 577, 626, 677, 730, 785,
842, 901, 962, 1025, 1090, 1157, 1226, 1297, 1370, 1445, 1522, 1601,
1682, 1765, 1850, 1937, 2026, 2117, 2210, 2305, 2402, 2501, .................

\(\D\sum_{k=0}^{\infty}\frac{k^2}{(k-0)!\times e}=1^2+1=2\)
\(\D\sum_{k=0}^{\infty}\frac{k^2}{(k-1)!\times e}=2^2+1=5\)
\(\D\sum_{k=0}^{\infty}\frac{k^2}{(k-2)!\times e}=3^2+1=10\)
\(\D\sum_{k=0}^{\infty}\frac{k^2}{(k-3)!\times e}=4^2+1=17\)
\(\D\sum_{k=0}^{\infty}\frac{k^2}{(k-4)!\times e}=5^2+1=26\)
\(\D\sum_{k=0}^{\infty}\frac{k^2}{(k-5)!\times e}=6^2+1=37\)
\(\D\sum_{k=0}^{\infty}\frac{k^2}{(k-6)!\times e}=7^2+1=50\)
\(\D\sum_{k=0}^{\infty}\frac{k^2}{(k-7)!\times e}=8^2+1=65\)
\(\D\sum_{k=0}^{\infty}\frac{k^2}{(k-8)!\times e}=9^2+1=82\)
         
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 5 天前 | 显示全部楼层
漂亮的公式,活跃的气氛。

\[\D\sum_{n=0}^{\infty}\frac{\pi^2}{n^2+\pi^2}=\frac{e^{\pi*\pi}+\pi*\pi*e^{\pi*\pi}+\pi*\pi*e^{-\pi*\pi}-e^{-\pi*\pi}}{e^{\pi*\pi}+e^{\pi*\pi}-e^{-\pi*\pi}-e^{-\pi*\pi}}\]

\[\D\sum_{n=1}^{\infty}\frac{\pi^2}{n^2+\pi^2}=\frac{e^{\pi*\pi}-\pi*\pi*e^{\pi*\pi}-\pi*\pi*e^{-\pi*\pi}-e^{-\pi*\pi}}{e^{-\pi*\pi}-e^{\pi*\pi}-e^{\pi*\pi}+e^{-\pi*\pi}}\]


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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