009C Sample Midterm 3

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 

Test if the following sequence ${a_{n}}$ converges or diverges.

If it converges, also find the limit of the sequence.

$a_{n}=\left({\frac {n-7}{n}}\right)^{\frac {1}{n}}$

 Problem 2 

For each the following series find the sum, if it converges.

If you think it diverges, explain why.

(a)  ${\frac {1}{2}}-{\frac {1}{2\cdot 3}}+{\frac {1}{2\cdot 3^{2}}}-{\frac {1}{2\cdot 3^{3}}}+{\frac {1}{2\cdot 3^{4}}}-{\frac {1}{2\cdot 3^{5}}}+\cdots$

(b)  $\sum _{n=1}^{\infty }\,{\frac {3}{(2n-1)(2n+1)}}$

 Problem 3 

Test if each the following series converges or diverges.

Give reasons and clearly state if you are using any standard test.

(a)  ${\displaystyle \sum _{n=1}^{\infty }}\,{\frac {n!}{(3n+1)!}}$

(b)  ${\displaystyle \sum _{n=2}^{\infty }}\,{\frac {\sqrt {n}}{n^{2}-3}}$

 Problem 4 

Test the series for convergence or divergence.

(a)  ${\displaystyle \sum _{n=1}^{\infty }}\,(-1)^{n}\sin {\frac {\pi }{n}}$

(b)  ${\displaystyle \sum _{n=1}^{\infty }}\,(-1)^{n}\cos {\frac {\pi }{n}}$

 Problem 5 

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

(a)  ${\displaystyle \sum _{n=0}^{\infty }}{\frac {(-1)^{n}x^{n}}{n+1}}$

(b)  ${\displaystyle \sum _{n=0}^{\infty }}{\frac {(x+1)^{n}}{n^{2}}}$