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Math Help - limit of the sequence

  1. #1
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    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),A(f),A(f),... when f (x)=1.for all x∈R
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  2. #2
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    Quote Originally Posted by makenqau View Post
    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),A(f),A(f),... when f (x)=1.for all x∈R
    So you start with f_0(x) = 1 and you construct a sequence of functions by the formula f_{n+1}(x) = 1+ \int_0^xf_n(t)\,dt:

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

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

    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 f_n(x), calculate 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→∞.
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