一、(每小题10分,共20分)设$displaystyle limlimits_{n o infty}x_{n}=a$设$displaystyle y_{n}=frac{x_{1}+2x_{2}+cdotcdotcdot+nx_{n}}{n(n+1)}$.证明:
1.设$a$是有限数,则$displaystyle limlimits_{n o infty}y_{n}=frac{a}{2}$.
2.若$a=+infty $,则$displaystyle limlimits_{n o infty}y_{n}=+infty$.
二、(每小题10分,共20分)设$f(x)$在$[0,+infty)$上单调递减且$displaystyle int_{0}^{+infty}f(x)dx$收敛.
1.证明:$displaystyle limlimits_{x o +infty}xf(x)=0$
2.若$x o +infty$时$displaystyle f(x) o 0$且$f'(x)$连续,证明:$displaystyle int_{0}^{+infty}xf'(x)dx$也收敛.
三、(每小题10分,共20分)
1.若对每个正整数$displaystyle n,u_{n}(x)$是$(0,1)$内的单调递减的函数,且$displaystyle limlimits_{x o 1-0}u_{n}(x)=1$,证明:
若$displaystyle sumlimits_{n=1}^{infty}a_{n}$收敛则$displaystyle limlimits_{x o 1-0} sumlimits_{n=1}^{infty}a_{n}u_{n}(x)=sumlimits_{n=1}^{infty} a_{n}$
2.证明:$ displaystyle limlimits_{x o 1-0}sumlimits_{n=1}^{infty}frac{(-1)^{n-1}x^{n}}{n(1+x^{n})}=frac{1}{2}ln 2$
四、(本题满分15分)设$y=f(x)$在$displaystyle [-sqrt{a^{2}+b^{2}+c^{2}},sqrt{a^{2}+b^{2}+c^{2}} ]$上连续,
证明:$ displaystyle iintlimits_{S} f(ax+by+cz)dS=2pi int_{-1}^{1}f(usqrt{a^{2}+b^{2}+c^{2}})du$.其中$S$是单位球面:$displaystyle x^{2}+y^{2}+z^{2}=1$.