Sichuan University 2005 years of Mathematical analysis of postgraduate examination questions

I. (15 points) try to limit $\displaystyle \lim\limits_{n\to \infty}\sum\limits_{k=1}^{n}\sin \frac{k}{n^{2}}$.

Ii. (15 points) known quantity column $\{x_{n}\}$ meet: All $n$ have $\displaystyle \left (a+\frac{1}{n}\right) ^{n+x_{n}}=e$ set up. $\displaystyle \ Lim\limits_{n\to \infty}x_{n}$.

Three, (the subject of 15 points) to calculate the double integral: $\displaystyle \iint_{d}e^{-(x+y) ^{2}}dxdy $, wherein $d$ is surrounded by $x+y=1,y=x,x=0$.

Iv. (15 points) if $\displaystyle-\infty <a<b<c<+\infty,f (x) $ on $[a,c]$ continuous, and $f (x) $ on $ (a,c) $ on differentiable guide. Verification: Presence $\xi \in (A,C) $ makes:

$$\frac{f (A)} {(a) (a-c)}+\frac{f (b)} {(b-c) (C-b)}+\frac{f (c)} {(c-a) (c-b)}=\frac{1}{2}f "(\XI) $$

Was founded

V. (15 points) for all $x\in (0,+\infty) $, Series $\displaystyle \sum\limits_{n=1}^{\infty}a_{n}x^{n}$ are convergent, and $\displaystyle \sum\ limits_{n=1}^{\infty}n!a_{n}$ Convergence.

Proof: $\displaystyle \int_{0}^{+\infty}\left (\sum\limits_{n=1}^{\infty}a_{n}x^{n}e^{-x}\right) Dx=\displaystyle \sum \limits_{n=1}^{\infty}n!a_{n}$.

Sichuan University 2005 years of Mathematical analysis of postgraduate examination questions