009C Sample Midterm 2

This is a sample, and is meant to represent the material usually covered in Math 9C for the midterm. An actual test may or may not be similar. Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

Evaluate:

a) $\lim _{n\rightarrow \infty }{\frac {1}{{\big (}{\frac {n-4}{n}}{\big )}^{n}}}$

b) $\sum _{n=1}^{\infty }{\frac {1}{2}}{\bigg (}{\frac {1}{4}}{\bigg )}^{n-1}$

 Problem 2 

Determine convergence or divergence:

$\sum _{n=1}^{\infty }{\frac {3^{n}}{n}}$

 Problem 3 

Determine convergence or divergence:

a) $\sum _{n=1}^{\infty }(-1)^{n}{\sqrt {\frac {1}{n}}}$

b) $\sum _{n=1}^{\infty }(-2)^{n}{\frac {n!}{n^{n}}}$

 Problem 4 

Find the radius of convergence and interval of convergence of the series.

a) $\sum _{n=0}^{\infty }n^{n}x^{n}$

b) $\sum _{n=0}^{\infty }{\frac {(x+1)^{n}}{\sqrt {n}}}$

 Problem 5 

The displacement from equilibrium of an object in harmonic motion on the end of a spring is:

$y={\frac {1}{3}}\cos(12t)-{\frac {1}{4}}\sin(12t)$

where $y$ is measured in feet and $t$ is the time in seconds. Determine the position and velocity of the object when $t={\frac {\pi }{8}}.$