Question

求助~~一道定积分的题~~

如图。。主要是第三小题不会做。。

Answer 1

(1)-|f(x)|≤f(x)≤|f(x)|
所以-∫|f(x)|dx≤∫f(x)dx≤∫|f(x)|dx
从而|∫f(x)dx|≤∫|f(x)|dx
(2)直接用等比数列求和公式就可以得到
(3)1/(1-x²)-(1+x²+...+x^(2n-2))=x^(2n)/(1-x²)
令f(x)=1/(1-x²)-(1+x²+...+x^(2n-2)),a=0,b=1/2
则|∫[0,1/2]f(x)dx|=|∫[0,1/2]1/(1-x²)dx-∫[0,1/2](1+x²+...+x^(2n-2))dx|
=|1/2ln3-(1/2+(1/2)^3/3+..+(1/2)^(2n-1)/(2n-1))|
记S[n]=(1/2+(1/2)^3/3+..+(1/2)^(2n-1)/(2n-1)),则
|∫[0,1/2]f(x)dx|=|1/2ln(3)-S[n]|
由于1/(1-x²)在[0,1/2]上是增函数,所以x^(2n)/(1-x²)<x^(2n)/(1-1/4)=4/3*x^(2n)
从而|∫[0,1/2]f(x)dx|=|1/2ln(3)-S[n]|≤∫[0,1/2]x^(2n)/(1-x²)dx
≤∫[0,1/2]4/3*x^(2n)dx=3/4*(1/2)^(2n+1)/(2n+1)
即|2S[n]-ln3|≤3/2*(1/2)^(2n+1)/(2n+1)
初步估算n<10,依次取n=9,8...
当n=8时3/2*(1/2)^(2n+1)/(2n+1)=0.67*10^(-6)<10^(-6)
所以q=2S[8]=810793999/738017280=1.098611131....