Sichuan University 2002 years of Mathematical analysis of postgraduate examination questions

One, (15 points) set $\displaystyle x_{1}>0,x_{n+1}=\frac{3 (1+x_{n})}{3+x_{n}} (N=1,2,\cdot \cdot \cdot) $, proving: $x _{n}$ has limits, and find the limit value.

Two, (15 points) set $y=f (x) $ in $\displaystyle [0,+\infty) $ consistent continuous, to any $\displaystyle x\in[0,1],\lim\limits_{n\to \infty} (X+n) =0$, ($ n$ is a positive integer),

Proof: $\displaystyle \lim\limits_{n\to \infty}f (x) =0$.

Three, (each small question 10 points, a total of 20 points) on the $[a,b]$, has $f "(x) >0$, proves:

1. For any $x_{0},x\in[a,b]$, there is $\displaystyle \displaystyle f (x) \ge f (x_{0}) +f ' (x-{0}) (X-x_{0}) $

2. For any $x_{1},x_{2},\cdot \cdot\cdot,x_{n}\in[a,b]$, there is $\displaystyle f\left (\frac{1}{n}\sum\limits_{i=1}^{n}x_{ I}\right) \le\frac{1}{n}\sum\limits_{i=1}^{n}f (X_{i}) $

Four, (15 points) set $y=f (x) $ in $[a,b]$ with continuous derivative and $f (a) =0$, proving: $$ M^{2}\le (b-a) \int_{a}^{b}[f ' (x)]^{2}dx$$

Wherein $\displaystyle M=\sup\limits_{a\le X\le b}\left| F (x) \right|$.

Five, (10 points) set $f (x, y) $ for $n$ times the homogeneous function, that is satisfied: to any $\displaystyle t>0,f (tx,ty) =t^{n}f (x, y) $, and $f$ can be micro, prove in $ (x, y) \not = (0,0) $ place

$$ x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf$$

Six, set $g (x) $ on $[0,1]$ continuous. Make $\displaystyle F_{n} (x) =g (x) x^{n}$, proving that: $\displaystyle \{f_{n} (x) \}$ is uniformly convergent on $[0,1]$.

VII, calculate integral $\displaystyle \int_{amb} (X^{2}-yz) dx+ (Y^{2}-XZ) dy+ (Z^{2}-XY) dz$ This integral is from point $ A (a,0,0) $ to Point $b (a,0,h) $ along Helix $\ Displaystyle X=a\cos \theta $,

$\displaystyle y=a\sin \theta $, $\displaystyle z=a\frac{h}{2\pi} \theta $ on the fetch.

Sichuan University 2002 years of Mathematical analysis of postgraduate examination questions