CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
21-256 Review Exam 1, Spring 2004

1. Determine if the following sequence converges or diverges. If it converges find the limit.

   
   

   $$a_n= \frac{2n^2+e^{-n}}{n^2+\cos n}$$
   
   

2. Determine if the following series converges or diverges. If it converges find the sum.

   
   

   $$\sum_{n=1}^{\infty} (-4)^{1-n} (\pi)^{n}$$
   
   

3. Determine if the following series converges or diverges.
   
   $$\sum_{n=1}^{\infty} \frac {\sqrt[4] {n^3+n^2+1} }{\sqrt[5]{1-n+n^9}}$$
   
   

4. Determine if the following series converges conditionally, converges absolutely, or diverges.
   
   $$\sum_{n=2}^{\infty} (-1)^{n-1} \frac{1}{n \ln n}$$
   
   

5. Determine if the following series converges conditionally, converges absolutely, or diverges.
   
   $$\sum_{n=1}^{\infty} \frac{\sin^n n}{n^n}$$
   
   

6. Determine if the following series converges conditionally, converges absolutely, or diverges.
   
   $$\sum_{n=0}^{\infty } \frac{(-e)^n}{n!}$$
   
   

7. Find the interval of convergence of the following series - you do not need to test endpoints.
   
   $$\sum_{n=1}^{\infty } n^2 4^n (x+3)^{2n}$$
   
   

8. Find a power series representation and interval of convergence of
   
   $$f(x)=\frac{ 2x^2}{4x+3} .$$
   
   

9. Find the Taylor series centered at $a=0$ for $f(x)=xe^{x^2}$. You may use the known series for $e^x$.