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Convergence of improper integrals theorems

  1. Oct 23, 2013 #1
    1. The problem statement, all variables and given/known data
    I'm trying to prove these two theorems
    a) if ## 0 \leq f(x) \leq g(x) ## for all x ## \geq 0 ## and ## \int_0^\infty g ## converges, then ## \int_0^\infty f ## converges
    b) if ## \int_0^\infty |f| ## converges then ## \int_0^\infty f ## converges.

    Obviously assuming f is integrable on every interval [0, N], N ## \geq 0##.

    2. Relevant equations


    3. The attempt at a solution
    For a).
    Let F be defined by ##F(x) = \int_0^x f ##. ##F## is bounded above by ## \int_0^\infty g ##. and since, ##0 \leq f(x)## for all x, F is non-decreasing. So ##\displaystyle\lim_{x\rightarrow \infty} F(x)## exists.

    For b)
    ##0 \leq f(x) + |f(x)| \leq 2|f(x)|## for all x. and ##\int_0^\infty 2|f| ## is convergent, so ##\int_0^\infty (f + |f|) ## is convergent by (a).
    The existence of ## \displaystyle\lim_{x\rightarrow \infty} \int_0^x |f|## and the existence of ##\displaystyle\lim_{x\rightarrow \infty} \int_0^x (f + |f|) = \displaystyle\lim_{x\rightarrow \infty}\left( \int_0^x f + \int_0^x |f| \right)##. implies the existence of ##\displaystyle\lim_{x\rightarrow \infty} \int_0^x f ##
     
  2. jcsd
  3. Oct 23, 2013 #2

    Ray Vickson

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    Homework Helper

    So, what is your question?
     
  4. Oct 23, 2013 #3
    Sorry. I just want to make sure that my proof is correct? I'm studying by myself.
     
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