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当$|x|<1$时，$\log (1+x)$的泰勒展开.解:是\begin{equation} \label{eq:11.13} x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+\cdots\end{equation}易得当$|x|<1$时，\beg... 阅读全文
posted @ 2012-11-12 23:51
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当$|x|<1$时，$\log (1+x)$的泰勒展开.解:是\begin{equation} \label{eq:11.13} x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+\cdots\end{equation}易得当$|x|<1$时，\beg... 阅读全文
posted @ 2012-11-12 23:51
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In order to study the convergence of $(1+\frac{1}{n})^n$ to$e$,consider the sequences \begin{equation} a_n=(1+\frac{1}{n})^n~~~\mbox{and}~~~b_n=(... 阅读全文
posted @ 2012-11-12 20:18
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In order to study the convergence of $(1+\frac{1}{n})^n$ to$e$,consider the sequences \begin{equation} a_n=(1+\frac{1}{n})^n~~~\mbox{and}~~~b_n=(... 阅读全文
posted @ 2012-11-12 20:18
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(Bernoulli's inequality;Jac.Bernoulli 1689,see 1744,Opera,p.380;Barrow1670,see 1860,Works,Lectio VII,XIII,p.224).By induction on $n$,provethat1.\begin... 阅读全文
posted @ 2012-11-12 15:04
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(Bernoulli's inequality;Jac.Bernoulli 1689,see 1744,Opera,p.380;Barrow1670,see 1860,Works,Lectio VII,XIII,p.224).By induction on $n$,provethat1.\begin... 阅读全文
posted @ 2012-11-12 15:04
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By using $2\cdot 4^3-5^3=3$,obtain the formula\begin{equation} \label{eq:12.38} \sqrt[3]{2}=\frac{5}{4}(1+\frac{1}{1\cdot 125}-\frac{2}{1\cdot 2\... 阅读全文
posted @ 2012-11-12 13:12
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By using $2\cdot 4^3-5^3=3$,obtain the formula\begin{equation} \label{eq:12.38} \sqrt[3]{2}=\frac{5}{4}(1+\frac{1}{1\cdot 125}-\frac{2}{1\cdot 2\... 阅读全文
posted @ 2012-11-12 13:12
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Verify the following formula(Euler 1755,Opera vol.X,p.280) by using$50=2\cdot 5^2=7^2+1$: \begin{equation} \label{eq:11.27} \sqrt{2}=\frac{7}{... 阅读全文
posted @ 2012-11-12 02:01
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Verify the following formula(Euler 1755,Opera vol.X,p.280) by using$50=2\cdot 5^2=7^2+1$: \begin{equation} \label{eq:11.27} \sqrt{2}=\frac{7}{... 阅读全文
posted @ 2012-11-12 02:01
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