2的自然对数

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ln2A002162)约为:

\ln 2 \approx 0.693147

使用对数公式

\log_b 2 = \frac{\ln 2}{\ln b}.

可以求出log2,它约为:(A007524

\log_{10} 2 \approx 0.301029995663981195

log210约为:

 \log_2 10=1/\log_{10} 2 \approx 3.321928095 *A020862)。
n ln n OEIS
2 0.693147180559945309417232121458 A002162
3 1.09861228866810969139524523692 A002391
4 1.38629436111989061883446424292 A016627
5 1.60943791243410037460075933323 A016628
6 1.79175946922805500081247735838 A016629
7 1.94591014905531330510535274344 A016630
8 2.07944154167983592825169636437 A016631
9 2.19722457733621938279049047384 A016632
10 2.30258509299404568401799145468 A002392

公式[编辑]

\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \sum_{n=0}^\infty \frac{1}{(2n+1)(2n+2)} = \ln 2.
\sum_{n=1}^\infty \frac{(-1)^n}{(n+1)(n+2)} = 2\ln 2 -1.
\sum_{n=1}^\infty \frac{1}{n(4n^2-1)} = 2\ln 2 -1.
\sum_{n=1}^\infty \frac{(-1)^n}{n(4n^2-1)} = \ln 2 -1.
\sum_{n=1}^\infty \frac{(-1)^n}{n(9n^2-1)} = 2\ln 2 -\frac{3}{2}.
\sum_{n=2}^\infty \frac{1}{2^n}[\zeta(n)-1] = \ln 2 -\frac{1}{2}.
\sum_{n=1}^\infty \frac{1}{2n+1}[\zeta(n)-1] = 1-\gamma-\frac{1}{2}\ln 2.
\sum_{n=1}^\infty \frac{1}{2^{2n}(2n+1)}\zeta(2n) = \frac{1}{2}(1-\ln 2).

\gamma欧拉-马歇罗尼常数\zeta黎曼ζ函數

\ln 2 = \sum_{k\ge 1} \frac{1}{k2^k}.
\ln 2 = \sum_{k\ge 1}(\frac{1}{3^k}+\frac{1}{4^k})\frac{1}{k}.
\ln 2 = \frac{2}{3}+\sum_{k\ge 1}(\frac{1}{2k}+\frac{1}{4k+1}+\frac{1}{8k+4}+\frac{1}{16k+12})\frac{1}{16^k}.贝利-波尔温-普劳夫公式
\ln 2 = \frac{2}{3} \sum_{k\ge 0} \frac{1}{(2k+1)9^k}.(基於反雙曲函數,可參見計算自然對數的級數。)

积分公式[编辑]

\int_0^1 \frac{dx}{1+x} = \ln 2.
\int_1^\infty \frac{dx}{(1+x^2)(1+x)^2} = \frac{1}{4}(1-\ln 2).
\int_0^\infty \frac{dx}{1+e^{nx}} = \frac{1}{n}\ln 2;
\int_0^\infty \frac{dx}{3+e^{nx}} = \frac{2}{3n}\ln 2.
\int_0^\infty [\frac{1}{e^x-1}-\frac{2}{e^{2x}-1}]=\ln 2.
\int_0^\infty e^{-x}\frac{1-e^{-x}}{x} dx= \ln 2.
\int_0^1 \ln\frac{x^2-1}{x\ln x}dx=-1+\ln 2+\gamma.
\int_0^{\pi/3} \tan x dx=2\int_0^{\pi/4} \tan x dx=\ln 2.
\int_{-\pi/4}^{\pi/4} \ln(\sin x+\cos x)dx=-\frac{\pi}{4}\ln 2.
\int_0^1 x^2\ln(1+x)dx=\frac{2}{3}\ln 2-\frac{5}{18}.
\int_0^1 x\ln(1+x)\ln(1-x)dx=\frac{1}{4}-\ln 2.
\int_0^1 x^3\ln(1+x)\ln(1-x)dx=\frac{13}{96}-\frac{2}{3}\ln 2.
\int_0^1 \frac{\ln x}{(1+x)^2}dx = -\ln 2.
\int_0^1 \frac{\ln(1+x)-x}{x^2}dx=1-2\ln2.
\int_0^1 \frac{dx}{x(1-\ln x)(1-2\ln x)} = \ln 2.
\int_1^\infty \frac{\ln\ln x}{x^3}dx = -\frac{1}{2}(\gamma+\ln 2).

\gamma欧拉-马歇罗尼常数

其他公式[编辑]

用皮尔斯展开式(A091846)表达ln2:

 \log 2 = \frac{1}{1} -\frac{1}{1\cdot 3}+\frac{1}{1\cdot 3\cdot 12} -\ldots.

恩格尔展开式A059180表达ln2:

\log 2 = \frac{1}{2}+\frac{1}{2\cdot 3}+\frac{1}{2\cdot3\cdot 7}+\frac{1}{2\cdot 3\cdot 7\cdot 9}+\ldots .

用余切展开式A081785表达ln2:

\log 2 = \cot(\arccot 0 -\arccot 1 +\arccot 5 -\arccot 55+\arccot 14187-\ldots).

其他對數[编辑]

範例[编辑]

10的自然對數[编辑]

參考文獻[编辑]

外部連結[编辑]

參見[编辑]