Exercises

1. Use any convergence test to determine if each of the following series converges or diverges and explain your answer.
   1. 
      
      $$\sum_{n=1}^{\infty} \frac{e^n}{3^n}$$
      
      

   2. 
      
      $$\sum_{n=1}^{\infty} \frac{1-\sin(n)\cos(n)}{n^2+1}$$
      
      

   3. 
      
      $$\sum_{n=1}^{\infty} \frac{1-\sin(n)\cos(n)}{n+1}$$
      
      

   4. 
      
      $$\sum_{n=1}^{\infty} \frac{\ln(n)}{\sqrt{n}}$$
      
      

   5. 
      
      $$\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2-\ln{n}}$$
      
      

   6. 
      
      $$\sum_{n=1}^{\infty} \frac{\sqrt{n^2+1}}{(n^3+1)}$$
      
      

   7. 
      
      $$\sum_{n=1}^{\infty} (\frac{n\pi}{\sqrt{9n^2+1}})^{n+1}$$
      
      

   8. 
      
      $$\sum_{n=1}^{\infty} \frac{n!}{n^n}$$
      
      

   9. 
      
      $$\sum_{n=1}^{\infty} (\sqrt{n+\sqrt{n}}-\sqrt{n})$$
      
      

   10. 
       
       $$\sum_{n=1}^{\infty} \frac{(\cos(n\pi)+2)}{n}$$