2018春高数下期中试题2(回忆版)

1

Determine which of the following series converges absolutely, converges or diverges. Use any method, and give reasons for your answers

1-1

$(1)\sum _{n=1}^{\mathrm{\infty }}\frac{x^n+4^n}{3^n+4^n}$

1-2

$(2)\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n(\ln n{)}^2}$

1-3

$(3)\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\sqrt[n]{n}}$

1-4

$(4)\sum _{n=1}^{\mathrm{\infty }}\frac{n!(n+1)!(n+2)!}{(3n)!}$

1-5

$(5)\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n(\sqrt{n^2+1}-n)$

2

Find the radius and interval of convergence of the series

2-1

$\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{x^n}{\sqrt{n^2+3}}$

2-2

For what values of $x$ does the series converge absolutely, or conditionally?

3

Find the Maclaurin series of the function

3-1

$f(x)=(x+1)e^x$

4

Use series to evaluate the limit

4-1

$\underset{x\to 0}{lim}\frac{\ln (1+x^2)}{1-\cos x}$

5

Find the length of astroid

5-1

$x={\cos }^{3}t,y={\sin }^{3}t,0⩽t⩽2\pi$

6

Find the area of the region bounded by the circle $r=2\sin \theta$ for $\frac{\pi }{4}⩽\theta ⩽\frac{\pi }{2}$

7

Find the first four terms of the binomial series for the function

7-1

$(1+x{)}^{\frac{1}{2}}$

8

Does the following sequence converge? If so, to what value?

$$x_1=1,x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n}$$