Thread: limit of the sequence

Let C(R) be the space of continuous functions of one real variable
and let
A:C(R)→C(R)
be the operator which maps the function f into the function A(f) defined by
the formula(Af)(x)=1+∫₀^{x}f(t)dt
. Find the limit of the sequence of
functions f,A(f),A²(f),A³(f),... when f (x)=1.for all x∈R

Originally Posted by makenqau

Let C(R) be the space of continuous functions of one real variable
and let
A:C(R)→C(R)
be the operator which maps the function f into the function A(f) defined by
the formula(Af)(x)=1+∫₀^{x}f(t)dt
. Find the limit of the sequence of
functions f,A(f),A²(f),A³(f),... when f (x)=1.for all x∈R

So you start with and you construct a sequence of functions by the formula :

,

,

.

If you can't already guess the formula for , calculate (and more if necessary). When you have guessed the formula, see if you can prove it by induction. Then look for the limit as n→∞.