Improper Integral

In this page, improper integrals will be defined and convergence tests

for improper integrals will be discussed.

• Definition of Improper Integral
  
  
  • First kind of improper integral.
    
    ( i ) If     exists for every     and     exists, then  
    
    converges and   .
    
    ( ii ) If     exists and for every     and     exists, then
    
      converges and   .
    
    ( iii ) If     and     are both convergent for some c, then
    
      converges and  
  
  
  Remarks :
  
  The above improper integrals (   or     or   )
  
  may be replaced by Riemann-Stieltijes integrals (   or     or
  
  )   or by Lebesque-Stieltjes integrals (   or
  
    or ).
  
  Example 1 : Show that
  
  ( i )     converges if   p > 1
  
  ( ii )     diverges if  
  
  Proof : ( 1 ) When   p > 1
  
            For any given ,
  
            .
  
            .   So,     converges.
  
  
            ( 2 ) When     , given  
  
           
  
           
  
           
  
            So,     diverges.
  
  Note:
  
  The above results are very useful for the convergence test of any improper integral.
  
  ( Especially,     can be used in the limit comparison test. )
  
  
  Example 2: Show that     diverges
  
  Proof: Given  
  
           
  
            The limit does not exist as  
  
  Example 3: Show that     converges if a > 0
  
  Proof: ( 1 ) Given   ,
  
           
  
           since a > 0
  
  
            ( 2 ) Given   ,
  
           
  
            Since a > 0
  
            So,
  
           
  
  Remark: if a > 0, then   diverges   So,   diverges, too.




• Convergence test for improper integral
  
  
  • Suppose     for all     and     exists for each   .
    
    Then     exists iff there exists a constant M > 0 such that  
    
    for every   .
  
  
  
  • Comparison Test.
    
    Suppose     for all   ,   and     and     exist
    
    If     converges, then     converges.
  
  
  
  
  • Limit Comparison Test.
    
    Suppose     and     for all    , and    
    
    and     exist for   .
    
    Let   ,
    
    ( i ) If     then both     and     converge or both diverge.
    
    ( ii ) If   Y = 0, then if     converges, then     converges.
    
    ( iii ) If   ,   then if     converges, then     converges.
  
  
  
  • Remark:
    
    ( 1 ) Since   ,       is increasing function of b. Therefore, if
    
        is bounded for all   b,   then     converges.
    
    ( 2 ) The first convergence test can be used to prove comparison test, since
    
      when     converges.
    
    ( 3 ) Limit Comparison Test.
    
          ( i ) If   ,   then given   ,   there exists   a < c   such that
    
          ,   whenever   .   i.e     and
    
            whenever   .
    
          Therefore, iff   converges,   then converges.
    
          ( ii ) If   Y = 0,   then given   ,   there exists   c > a   such that  
    
          whenever   .
    
          Therefore, if     converges, then     converges.
    
          ( iii ) If   ,   then given   M > 0,   there exists   c > a   such that  
    
          whenever   .
    
          Therefore, if     converges, then     converges.
    
    
          Example: Test the improper integral for convergence
    
          ( i )  
    
          Compare     with a     by limit comparison test. It is easily
    
          to show when p = 1,  
    
          Therefore,     diverges, since     diverges.  
    
          i.e   diverges.
    
          ( ii )  
    
          ( a ) If s > 1, then converges ( By definition )
    
          ( b ) If    , then diverges (By definition)
    
    
          ( iii )     converges by limit comparison test using   .
    
          Therefore,     converges.