# Thread: limit of the sequence

1. ## 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

2. 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 $\displaystyle f_0(x) = 1$ and you construct a sequence of functions by the formula $\displaystyle f_{n+1}(x) = 1+ \int_0^xf_n(t)\,dt$:

$\displaystyle f_1(x) = 1 + \int_0^x1\,dt = 1+x$,

$\displaystyle f_2(x) = 1 + \int_0^x(1+t)\,dt = 1+x + \tfrac12x^2$,

$\displaystyle f_3(x) = 1 + \int_0^x(1+t+\tfrac12x^2)\,dt = 1+x + \tfrac12x^2 + \tfrac16x^3$.

If you can't already guess the formula for $\displaystyle f_n(x)$, calculate $\displaystyle f_4(x)$ (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→∞.