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)  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 

If $\sum _{n=0}^{\infty }c_{n}x^{n}$ converges, does it follow that the following series converges?

(a) $\sum _{n=0}^{\infty }c_{n}{\bigg (}{\frac {x}{2}}{\bigg )}^{n}$

(b) $\sum _{n=0}^{\infty }c_{n}(-x)^{n}$