• Prove the following relationship:

(1-p)² sum _{s = 1 to infinity}  ( s²  p^s ) =
(1-p)² (p d/dp)²  sum _{s = 1 to infinity}  ( p^s )
 
 
 

Given that:

S= (1-p)² sum _{s = 1 to infinity}  ( s²  p^s )

please show:

S = (1 + p) /  (1 - p)        (p < p_c)
 
 
 

• From   g(r) = p^r   and g(r) = exp(-r/xi), derive: xi  ~ 1/(p_c - p)