Part 3

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

   Solution: Converges absolutely. Take absolute value. Compare to $b_n=\frac{\pi^n}{n!}$, series which converges by the ratio test.

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

   Solution: Converges by the ratio test.

3. Determine if the following series converges or diverges.
   
   $$\sum_{n=1}^{\infty} \arctan \left( \frac{1}{n} \right)$$
   
   

   Solution: Diverges by L.C.T, with $b_n=\frac{1}{n}$.

4. Approximate the sum of the following series to within $0.1$
   
   $$\sum_{n=1}^{\infty} \frac{1}{n+4^n}$$
   
   

Solution: Use comparison test, $b_n=\frac{1}{4^n}$. Need $n=1$. $s \approx 0.2$.