感谢 elim 多年来的鼓励！
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\[\D\lim_{n\to\infty}\bigg(\frac{2n+a}{2n-a}\bigg)^n=e^a\]
谢谢 cz1 !
\[\displaystyle\lim_{n\to\infty}\bigg(\frac{2n+\pi}{2n-\pi}\bigg)^n=e^{\pi}\]
不错不错
猜想。
\(\D a(n)=\bigg\lceil\frac{n}{\sqrt{e}}\bigg\rceil=n,\ \ n=1,2,3,4,5,6,7,8,9, ......\)
张狂一点，教科书应该这样写！
\(\big(\frac{n+a}{n}\big)^n<\big(\frac{2n^2+an+1}{2n^2-an+1}\big)^n< e^a<\big(\frac{2n^2+an-1}{2n^2-an-1}\big)^n<\big(\frac{n}{n-a}\big)^n\)
\[\D e=\sum_{n=0}^{\infty}\frac{3n^2+3n+1}{10n!}=\sum_{n=0}^{\infty}\frac{3n^2-3n+1}{4n!}=\cdots\cdots\]
伟大的“e”!!!
\[\displaystyle e=\sum_{n=0}^{\infty}\frac{3n^2-3n+1}{4n!}=\sum_{n=0}^{\infty}\frac{4n^2-3n+4}{9n!}=\sum_{n=0}^{\infty}\frac{3n^2+3n+1}{10n!}\]
\[\displaystyle e=\sum_{n=0}^{\infty}\frac{n^3+4n-1}{8n!}=\sum_{n=0}^{\infty}\frac{n^5-3n^4+n^3}{12n!}=\sum_{n=0}^{\infty}\frac{n^5+n^3-9}{48n!}\]
伟大的“e”!!!
\[\displaystyle e=\sum_{n=0}^{\infty}\frac{2n-1}{n!}=\sum_{n=0}^{\infty}\frac{3n^2-5n}{n!}=\sum_{n=0}^{\infty}\frac{4n^3-9n^2}{2n!}\]
\(\displaystyle e=\sum_{n=0}^{\infty}\frac{n^4-(n-1)^4+n^3-(n+1)^3}{n!}\)
\(\displaystyle e=\sum_{n=0}^{\infty}\frac{n^5-(n-1)^5+n^4-(n+1)^4}{4n!}\)
\(\displaystyle e=\sum_{n=0}^{\infty}\frac{n^6-(n-1)^6+n^5-(n+1)^5}{11n!}\)
\(\displaystyle e=\sum_{n=0}^{\infty}\frac{n^7-(n-1)^7+n^6-(n+1)^6}{41n!}\)
\(\displaystyle e=\sum_{n=0}^{\infty}\frac{n^8-(n-1)^8+n^7-(n+1)^7}{162n!}\)
\(\displaystyle e=\sum_{n=0}^{\infty}\frac{n^9-(n-1)^9+n^8-(n+1)^8}{715n!}\)
......
怎么化简？
可以这样化简。
\(\displaystyle e=\sum_{n=0}^{\infty}\frac{n^{k+1}-(n-1)^{k+1}+n^{k}-(n+1)^{k}}{(n^{k-1}-(n-1)^{k-1})n!}\)
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}}的连分 ...
接12楼。
{1, 1, 1, 1, 05, 1, 1, 09, 1, 1, 13, 1, 1, 17, 1, 1, 021, 1, 1, 025, 1, 1, 029, 1, 1, 033, 1, 1, 037, 1, 1, 041, 1, 1, 045, 1, 1, 049, 1, 1, 053, 1, 1, 057, 1, 1, 061, 1, 1, 065},
{1, 2, 1, 1, 08, 1, 1, 14, 1, 1, 20, 1, 1, 26, 1, 1, 032, 1, 1, 038, 1, 1, 044, 1, 1, 050, 1, 1, 056, 1, 1, 062, 1, 1, 068, 1, 1, 074, 1, 1, 080, 1, 1, 086, 1, 1, 092, 1, 1, 098},
{1, 3, 1, 1, 11, 1, 1, 19, 1, 1, 27, 1, 1, 35, 1, 1, 043, 1, 1, 051, 1, 1, 059, 1, 1, 067, 1, 1, 075, 1, 1, 083, 1, 1, 091, 1, 1, 099, 1, 1, 107, 1, 1, 115, 1, 1, 123, 1, 1, 131},
{1, 4, 1, 1, 14, 1, 1, 24, 1, 1, 34, 1, 1, 44, 1, 1, 054, 1, 1, 064, 1, 1, 074, 1, 1, 084, 1, 1, 094, 1, 1, 104, 1, 1, 114, 1, 1, 124, 1, 1, 134, 1, 1, 144, 1, 1, 154, 1, 1, 164},
{1, 5, 1, 1, 17, 1, 1, 29, 1, 1, 41, 1, 1, 53, 1, 1, 065, 1, 1, 077, 1, 1, 089, 1, 1, 101, 1, 1, 113, 1, 1, 125, 1, 1, 137, 1, 1, 149, 1, 1, 161, 1, 1, 173, 1, 1, 185, 1, 1, 197},
{1, 6, 1, 1, 20, 1, 1, 34, 1, 1, 48, 1, 1, 62, 1, 1, 076, 1, 1, 090, 1, 1, 104, 1, 1, 118, 1, 1, 132, 1, 1, 146, 1, 1, 160, 1, 1, 174, 1, 1, 188, 1, 1, 202, 1, 1, 216, 1, 1, 230},
{1, 7, 1, 1, 23, 1, 1, 39, 1, 1, 55, 1, 1, 71, 1, 1, 087, 1, 1, 103, 1, 1, 119, 1, 1, 135, 1, 1, 151, 1, 1, 167, 1, 1, 183, 1, 1, 199, 1, 1, 215, 1, 1, 231, 1, 1, 247, 1, 1, 263},
{1, 8, 1, 1, 26, 1, 1, 44, 1, 1, 62, 1, 1, 80, 1, 1, 098, 1, 1, 116, 1, 1, 134, 1, 1, 152, 1, 1, 170, 1, 1, 188, 1, 1, 206, 1, 1, 224, 1, 1, 242, 1, 1, 260, 1, 1, 278, 1, 1, 296},
{1, 9, 1, 1, 29, 1, 1, 49, 1, 1, 69, 1, 1, 89, 1, 1, 109, 1, 1, 129, 1, 1, 149, 1, 1, 169, 1, 1, 189, 1, 1, 209, 1, 1, 229, 1, 1, 249, 1, 1, 269, 1, 1, 289, 1, 1, 309, 1, 1, 329}
......
Table, 50], {a, 2, 10}]