习题

mix

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• \(\int \frac{a}{b+\sin^2(x)}dx = \int \frac{-a}{b(1+\cot^2(x))+1}d\cot(x)\)

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• \(\arctan x = \sum\limits_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{2n+1}\)

• \(\int_0^{+\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2}\)

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• \(x^2+y^2\leq (|x|+|y|)^2\)

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• 若 \(\sum\limits_{i=1}^n p_i=1,0<p_i<1\), 对任意的实数 \(x_1, x_2, \cdots, x_n\), 证明:

  • \[\sum_{i=1}^n p_i\left(x_i-\ln p_i\right) \leq \ln \left(\sum_{i=1}^n \mathrm{e}^{x_i}\right)\]

• Proof. 由于 \(\ln x\) 是下凸函数, 于是由 Jensen 不等式知

  • \[\sum_{i=1}^n p_i\left(x_i-\ln p_i\right)=\sum_{i=1}^n p_i \ln \frac{\mathrm{e}^{x_i}}{p_i} \leq \ln \left(\sum_{i=1}^n p_i \cdot \frac{\mathrm{e}^{x_i}}{p_i}\right) \ln \left(\sum_{i=1}^n \mathrm{e}^{x_i}\right)\]

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• \(\int_0^{\frac{\pi}{2}}f(\sin x) dx =\int_0^{\frac{\pi}{2}}f(\cos x) dx\)

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09 数项级数

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• \(\sqrt[n]{n} - 1 > e^{\frac{1}{n}} - 1 \sim \frac{1}{n},n\to\infty\)
  • \(\sum(\sqrt[n]{n}-1)\) 发散

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• \(\frac{1}{n+1} < \int_n^{n+1} \frac{dx}{x} < \frac{1}{n}\)

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15 含参变量积分

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• \(\int_0^{\frac{\pi}{2}} \ln(\cos(x))dx = -\frac{\pi}{2}\ln(2)\)

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