Practice Exam 2

1. Explain what occurs when you aaply Newton's Method to find the root of $f(x)=\sqrt[3]{x}$ with $x_1=1$.

2. How many terms of a partial sum are required so that the approximation of the following series is accurate to within $0.05$?
   
   $$\sum_{n=0}^{\infty} \frac{ 1}{2^n n!}$$
   
   

3. How many terms of a partial sum are required so that the approximation of the following series is accurate to within $0.05$?
   
   $$\sum_{n=1}^{\infty} \frac{ (-1)^{n-1}}{n^2}$$
   
   

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 or diverges.
   
   $$\sum_{n=1}^{\infty} \ln \left( \frac{2n}{1+n} \right)$$
   
   

6. Determine if the following series converges conditionally, converges absolutely, or diverges.
   
   $$\sum_{n=1}^{\infty} \frac{(-\pi)^n}{n+n!}$$
   
   

7. Determine if the following series converges or diverges.
   
   $$\sum_{n=1}^{\infty} \frac {\sqrt {n+1} }{1-n+n^2}$$
   
   

8. Determine if the following series converges or diverges.
   
   $$\sum_{n=0}^{\infty} \frac{1 \cdot 4 \cdot 7 \cdots (3n+1)}{2 \cdot 5 \cdot 8 \cdots (3n+2)}$$