I. (10 points) for the limit $\displaystyle \lim\limits_{n\to \infty}\left (n-\frac{1}{e^{\frac{1}{n}}-1}\right) $
Ii. (15 points) Set function $f (x) $ in $[0,1]$ Ober guide, meet $\displaystyle | F ' (x) |\le 1,f (x) $ within the range $ (0,1) $ to take the maximum value $\displaystyle \frac{1}{4}$,
Proof: $| F (0) |+|f (1) | \le 1.$
Three, (15 points) Set function $f (x) $ on $[0,1]$ continuous, and $\displaystyle F (0) =f (1) =0$. Proof: $$ \int _{0}^{1} \left| f (x) F ' (x) \right| Dx\le \frac{1}{4}\int_{0}^{1}[f ' (x)]^{2}dx.$$
Four, (20 points) proving function item progression $\displaystyle \sum\limits_{n=1}^{\infty}\left ( -1\right) ^{n+1}\frac{1}{n^{x}}$ in $ (0,+\infty) $ On inconsistent convergence,
But in $ (0,+\infty) $ there is a continuous derivative.
V. (15 points) calculate the surface area of $\displaystyle \iint\limits_{s}\frac{xdydz+ydzdx+zdxdy}{\left (x^{2}+y^{2}+z^{2}\right) ^{\frac{ 3}{2}}}$. Where $s$ is an ellipsoid: $\displaystyle \frac{x^{2}}{a^2}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 (Z\GE0) $ on the upper side.
Sichuan University 2006 years of Mathematical analysis of postgraduate examination questions