Comparison Tests For Convergence Of Improper Integrals

Comparison Tests For Convergence:

Comparison Test 1 (Comparison Of                                    Two Integrals) :

           If f and g be two positive functions such that f(x)≤g(x) , for all x in [a,b] , then

           b                                  b

    (1)  ∫ f dx converges , if ∫ g dx converges

          a                                   a

and   b                               b

   (2)  ∫ g dx diverges , if ∫ f dx diverges .

         a                                a

Comparison Test (2)  (Limit Form) :

        If f and g are two positive functions in [a,b] such that    lim    f(x)/g(x)  = l ,

                            x-->a+0

   Where l is a non _ zero finite number , 

                                          b                 b

then the two integrals ∫ f dx and ∫ g dx 

                                         a                 a

converges and diverges together at a .

Theorem Related To Convergence Of Improper Integral :

Theorem 1 :

               A necessary and sufficient condition for the convergence of the improper integral
              b
             ∫ f dx at a, where f is positive in [a,b]

             a

is that there exists a positive number M , independent of λ , such that
             b
             ∫ f dx < M , 0<λ<b-a
          a+λ

Proof :

         We know that the improper integral 

    b                                                             b

   ∫ f dx converges at a if for 0<λ<b-a ∫ f dx 

   a                                                            a+λ

tends to a finite limit as λ-->0⁺

   Since f is positive in [a,b] the positive 

                           b

function of λ , ∫ f dx is monotone increasing

                         a+λ

as λ decreases and will therefore tend to a finite limit iff it is bounded above i.e ∃ a positive number M independent of λ such that 

              b

              ∫ f dx < M ,  0<λ<b-a

            a+λ

Hence the proof .