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10 的自然对数

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10 的自然对数的十进制展开式由下式给出

ln10=2.302585092994045684...
(1)

(OEIS A002392)。

它也由 BBP 型公式给出

ln10=1/(32)sum_(k=0)^(infty)1/((-64)^k)((64)/(12k+1)+(16)/(12k+2)+8/(12k+4)-(16)/(12k+5)+8/(12k+6)+(12)/(12k+7)-2/(12k+8)+4/(12k+9)+1/(12k+10)-3/(12k+11)-1/(12k+12))
(2)
=1/2sum_(k=0)^(infty)1/((-4)^k)(6/(4k+1)-3/(4k+3)-1/(4k+4))
(3)
=1/(16)sum_(k=0)^(infty)1/(16^k)((24)/(4k+1)+(20)/(4k+2)+6/(4k+3)+1/(4k+4))
(4)
=1/8sum_(k=0)^(infty)1/(16^k)((16)/(8k+1)+8/(8k+2)-8/(8k+3)+4/(8k+4)-4/(8k+5)+2/(8k+6)+2/(8k+7)+1/(8k+8))
(5)
=2/9sum_(k=0)^(infty)1/(81^k)(9/(4k+1)+2/(4k+2)+1/(4k+3))
(6)
=2/(729)sum_(k=0)^(infty)1/(6561^k)((729)/(8k+1)+(162)/(8k+2)+(81)/(8k+3)+9/(8k+5)+2/(8k+6)+1/(8k+7))
(7)

(E. W. Weisstein，2008 年 8 月 28 日)。

另请参阅

BBP 型公式, 自然对数, 2 的自然对数, 10 的自然对数连分数, 10 的自然对数数字

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参考文献

Sloane, N. J. A. Sequence A002392/M0394 in "The On-Line Encyclopedia of Integer Sequences."Mansell, W. E. Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8. Cambridge, England: Cambridge University Press, p. 2, 1964.Uhler, H. S. "Recalculation and Extension of the Modulus and of the Logarithms of 2, 3, 5, 7 and 17." Proc. Nat. Acad. Sci. U. S. A. 26, 205-212, 1940.

请引用为

Weisstein, Eric W. "Natural Logarithm of 10." 来自 MathWorld——Wolfram Web 资源。 https://mathworld.net.cn/NaturalLogarithmof10.html

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