证明:In=∫0π2(sin(nx)sin(x))2dx=nπ2 当n=1时 I1=∫0π2(sin(x)sin(x))2dx=∫0π21dx=π2原式成立 若n=k 假设条件Ik=∫0π2(sin(kx)sin(x))2dx=kπ2......(1)成立 那么 当n=k+1 Ik+1=∫0π2(sin((k+1)x)sin(x))2dx=∫0π2(sin(kx)cos(x)+co
证明: \int_{0}^{\pi/2 } (\frac{\sin nx}{\sin x} )^{2} =\frac{n\pi }{2} . 此题摘自谢惠民数学分析习题讲义10.4.6练习题第十题,看到知乎上有费耶尔积分的复变证明,写个实变的证明。 证:设 I_{n}= \int_{0}^{\pi/2 } (\frac{\sin nx}{\sin x} )^{2} =\frac{n\pi }...
解 当n→∞∑m=0n−1(1/n)f(m/n)∫01g(t)dt=(∫01f(x)dx)(∫01g(x)dx)又|∫m/n...