Problem 65b

Step 1.

First time, we remove   of the original square.

Second time, we remove  of the eight squares remaining from the first time i.e

   

 

Third time, we remove    of the sixty-four squares remaining from the second time i.e

 

nth the we will remove  of the  squares remaining after the th removal i.e.,  

 

 

 

Now we sum the areas of the removed squares:

    Sum of removed areas

                                   

 

This is a geometric series with  , and   since  , it's convergent and we car compute its sum. as follows

 

 

 

Hence the removed area will be   .