(* Part (a) *)
Series[ (((1-(1/2)B))/(1-B)),{B,0,10}]
Series[ (((1-B))/(1-(1/2)B)),{B,0,10}]

(* Part (b) *)
Series[ (((1-(1/5)B))/(1-B)),{B,0,10}]
Series[ (((1-B))/(1-(1/5)B)),{B,0,10}]

(* Part (c) *)
Series[ (1/((1-B)(1-(1/2)B))),{B,0,10}]
Series[ (1-B)(1-(1/2)B),{B,0,10}]

(* Part (d) *)
Series[ (1/((1-B)(1-(1/5)B))),{B,0,10}]
Series[ (1-B)(1-(1/5)B),{B,0,10}]

(* make sure that our algebra in the text is correct *)
Table[(1/4)((((5^(j+1))-1))/(5^j)),{j,0,10}]

(* Part (e) *)
Series[ ((1-(1/2)B)/((1-B)(1-(1/5)B))),{B,0,10}]
Series[ (1-B)(((1-(1/5)B))/(1-(1/2)B)),{B,0,10}]