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墨卡托級數.
在數學內,墨卡托級數(Mercator series)或者牛頓-墨卡托級數(Newton–Mercator series)是一個自然對數的泰勒級數:
![{\displaystyle \ln(1+x)\;=\;x\,-\,{\frac {x^{2)){2))\,+\,{\frac {x^{3)){3))\,-\,{\frac {x^{4)){4))\,+\,\cdots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7efed2560a7c78dfba3a7e85ec9af57c97d458)
使用大寫sigma表示則為
![{\displaystyle \ln(1+x)\;=\;\sum _{n=1}^{\infty }{\frac {(-1)^{n+1)){n))x^{n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e7ab31275628e74fd3a456e12225af5716bdcb8)
當 −1 < x ≤ 1時,此級數收斂於自然對數(加了1)。
歷史
這級數被尼古拉斯·墨卡托,牛頓和Gregory Saint-Vincent分別獨立發現。首先被墨卡托出版於其1668年時的著作Logarithmo-technica。
推導
這級數可以由泰勒公式導出,藉由不斷地計算第n次ln x在x = 1時的微分,一開始是
![{\displaystyle {\frac {d}{dx))\ln x={\frac {1}{x)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/908d58e860196c61db8f688876e2671bf1c812b0)
或者,我們可以從有限的等比數列開始(t ≠ −1)
![{\displaystyle 1-t+t^{2}-\cdots +(-t)^{n-1}={\frac {1-(-t)^{n)){1+t))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86b9459e0d1c0661ccac7b004f15f859c244ebf8)
這可以導出
![{\displaystyle {\frac {1}{1+t))=1-t+t^{2}-\cdots +(-t)^{n-1}+{\frac {(-t)^{n)){1+t)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ea5dfdd1bac3dd68214e3295393ca4f667694a)
然後得到
![{\displaystyle \int _{0}^{x}{\frac {dt}{1+t))=\int _{0}^{x}\left(1-t+t^{2}-\cdots +(-t)^{n-1}+{\frac {(-t)^{n)){1+t))\right)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2189b65a8a7787cdfe8102b4140bc72f33758bd)
接著逐項積分,
![{\displaystyle \ln(1+x)=x-{\frac {x^{2)){2))+{\frac {x^{3)){3))-\cdots +(-1)^{n-1}{\frac {x^{n)){n))+(-1)^{n}\int _{0}^{x}{\frac {t^{n)){1+t))\,dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4b6b971316c8538161754971fa8ebcd57564df)
若−1 < x ≤ 1,餘項會在
時趨近於零。
這個表示法可以重複積分k次,得到
![{\displaystyle -xA_{k}(x)+B_{k}(x)\ln(1+x)=\sum _{n=1}^{\infty }(-1)^{n-1}{\frac {x^{n+k)){n(n+1)\cdots (n+k))),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e80cafef1c49c4ea7115d773db7dfc5f3ebe142)
這裡的
![{\displaystyle A_{k}(x)={\frac {1}{k!))\sum _{m=0}^{k}{k \choose m}x^{m}\sum _{l=1}^{k-m}{\frac {(-x)^{l-1)){l))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/198ed584580a1acdc903abf0e8488c536c9dff31)
和
![{\displaystyle B_{k}(x)={\frac {1}{k!))(1+x)^{k))](https://wikimedia.org/api/rest_v1/media/math/render/svg/f46e720ea394fd42dfd72575655d9ad22f8a83c7)
都是x的多項式。[1]
特例
令墨卡托級數裡面的x = 1,則我們會得到交錯調和級數
![{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1)){k))=\ln 2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efe9ec8790ba92136c5f414e3c11f0e30ae3db8b)
複數級數
下面的複數冪級數
![{\displaystyle z\,-\,{\frac {z^{2)){2))\,+\,{\frac {z^{3)){3))\,-\,{\frac {z^{4)){4))\,+\,\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5df3f23c01535b11cb24918729b8faa35bc9413d)
是ln(1 + z)的泰勒級數,這裡ln代表複對數(complex logarithm)的主要分支(principal branch)。這個級數收斂於一個開放的單位圓盤 |z| < 1 以及圓 |z| = 1 , z = -1除外 (根據阿貝爾判別法),而且這裡的收斂對每個半徑小於一的圓盤是一致的 。
參考資料
- ^ Medina, Luis A.; Moll, Victor H.; Rowland, Eric S. Iterated primitives of logarithmic powers. 2009. arXiv:0911.1325
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