Exercises

Sessions

 

Exercise 1.   Let  and  be integers   Consider the  power series

 

and put  

a)  By induction on  show that

 +...+ 

and, by induction on deduce the expansion

     

where    denotes the binomial coefficient  

b)  Using show that

      =    

(This generalises (1) in the case where )

Exercise 2.  Determine the radius of convergence of each of the series:

a)     ()            b)   ( is a positive integer)

c)  with   for  n≥0, where a and b are real numbers such that .

Exercise 3.  a) Prove the Unicity Theorem:

If  then

Exercise 4.  Prove the Uniqueness Theorem:

Suppose and  are power series with the same radius of convergence R.

If =  then for  n=0,1,2,…

Exercise 5.  Consider the power series  and    Put

    

Show the following relations

 and if  

Exercise 6.   Let  C  ,  not being an integer   What is the radius of

convergence of

 

Show that its sum  for  satisfies the differential equation

Exercise 7.   Let  be a  power series with   Put

 for  and  

Show that

i)   

ii)  = for  

Exercise 8.   Let  be a  power series with

 for  where  and  are two given real numbers.

a)  Show that for  we have  where

 and deduce that the radius of convergence  is nonzero.

b)  Show that  for  and deduce that for   we have

c)      Let and  be two roots of   Using the decomposition of the right-hand side of (1), find an expression of  using and  and deduce that

 (Note that if  then 

Exercise 9.  Prove the following inequality