The soldier is 0.8944 miles away from his camp towards East-
North. 

It is obvious that he is in East-North direction. 

Distance travelled in North and South directions 
= 1/2 - 1/8 + 1/32 - 1/128 + 1/512 - 1/2048 + and so on... 
(a geometric series with r = (-1/4) ) 

   (1/2) * ( 1 - (-1/4)n ) 
= --------------------------- 
         ( 1 - (-1/4) ) 

= 1 / ( 2 * ( 1 - (-1/4) ) ) 
= 2/5 

Similarly in East and West directions 
= 1 - 1/4 + 1/16 - 1/64 + 1/256 - and so on... (a geometric 
series with r = (-1/4) ) 

   (1) * ( 1 - (-1/4)n ) 
= --------------------------- 
         ( 1 - (-1/4) ) 

= 1 / ( ( 1- (-1/4) ) 
= 4/5 

So the soldier is 4/5 miles away towards East and 2/5 miles 
away towards North. So using right angled triangle, soldier 
is 0.8944 miles away from his camp.