help finding the sum of infinite series

Question

 

how do I find the sum of the following infinite series:

 

 

1.  1/2 + 1 + 2 + 4......

 

2.  9 - 3 + 1 - 1/3 + 1/9 .....

 

 

thanks

Answer 1

1. 1/2 + 1 + 2 + 4......diverges.  The terms keep getting larger.

 

2. 9 - 3 + 1 - 1/3 + 1/9 ..... converges.  It's an alternating series whose terms go to 0.  It is a geometric series with ratio -1/3.

The sum is 9/(1-(-1/3))=9/(4/3)=27/4=6 3/4