Simple example

 

• let
  
  

• then we can express the ratio
  
  

• where we have denoted

  
  
  

• the idea is to try to express the sum as

   

• where is a polynomial in ; note that

  
   
  

• substituting (2.1) into (2.2), we obtain a recurrence relation for :

  
   
  

• to solve the recurrence relation, we need to know the degree of the polynomial

• we can rewrite (2.3) as

   

• introducing

  
  
  

  and substituting this formula into (2.4), we obtain

  
  
  

• from which it follows that (if , then the previous equation implies that )

• therefore, is a polynomial of at most degree two

  
  
  

• the solution of (2.3) is then
  
  where is an arbitrary real parameter

• the value of is obtained from the initial condition , which gives and hence,
  
  

• the final solution is