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					Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS

    THEOREM: (Integral Test)
     f (x) a continuous, positive, decreasing function on [1, inf)    f ( n )  an
        

        an Convergent
                                                   

       n 1
                                               1
                                                       f ( x ) dx Convergent

        

        an Divergent
                                                   

       n 1
                                               1
                                                       f ( x ) dx Dinvergent


     Example:                                 Example:
     Test the series for                      Test the series for
     convergence or divergence.               convergence or divergence.
                                                        
                 1
             n2  1
            n 1
                                                         n
                                                             ln n
                                                        n 1
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS

    THEOREM: (Integral Test)
     f (x) a continuous, positive, decreasing function on [1, inf)      f ( n )  an
        

        an Convergent
                                                    

       n 1
                                                1
                                                         f ( x ) dx Convergent

        

        an Divergent
                                                    

       n 1
                                                1
                                                         f ( x ) dx Dinvergent


                    REMARK:
                    When we use the Integral Test, it is not
                    necessary to start the series or the integral
                    at n = 1 . For instance, in testing the series
                       
                           1
                       (n  3) 2
                                                           dx
                      n4
                                                    4   (x  3) 2
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS

       THEOREM: (Integral Test)
       f (x) a continuous, positive, decreasing function on [1, inf)      f ( n )  an
           

           an Convergent
                                                     

          n 1
                                                 1
                                                         f ( x ) dx Convergent

           

           an Divergent
                                                     

          n 1
                                                 1
                                                         f ( x ) dx Dinvergent


REMARK:                                                  REMARK:
When we use the Integral Test, it is not              Also, it is not necessary that f(x)
necessary to start the series or the integral         be always decreasing. What is
at n = 1 . For instance, in testing the series        important is that f(x) be ultimately
                                                      decreasing, that is, decreasing for
   
       1
   (n  3) 2
                                       dx            larger than some number N.

  n4
                             4      (x  3) 2
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS

      THEOREM: (Integral Test)
      f (x) a continuous, positive, decreasing function on [1, inf)       f ( n )  an
          

          an Convergent
                                                    

         n 1
                                                1
                                                        f ( x ) dx Convergent

          

          an Divergent
                                                    

         n 1
                                                1
                                                        f ( x ) dx Dinvergent

                                                Special Series:
                                                                               
 Example:                                                                    ar     n 1

                                                1) Geometric Series         n 1
 For what values of p is the series
                                                                           
 convergent?
                    
                                                2) Harmonic Series              1
                                                                                 n
                        1
                    np
                                                                          n 1

                                                3) Telescoping Series          
                   n 1
                                                                 
                                                                             (b b
                                                                            n 1
                                                                                     n      n 1   )
                                                                      1
                                                4) p-series       np
                                                                 n 1
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS

      THEOREM: (Integral Test)
      f (x) a continuous, positive, decreasing function on [1, inf)    f ( n )  an
          

          an Convergent
                                                    

         n 1
                                                   1
                                                        f ( x ) dx Convergent

          

          an Divergent
                                                    

         n 1
                                                   1
                                                        f ( x ) dx Dinvergent


 Example:
                                                P Series:
 For what values of p is the series
 convergent?                                    
                                                    1 convg                 p 1
                                                n p   divg
                    
                        1
                    np
                   n 1
                                               n 1                         p 1
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS

  Example:
                                         P Series:
  For what values of p is the series
  convergent?                           
                                            1 convg               p 1
                                        n p   divg
                       
                         1
                     np
                    n 1
                                       n 1                       p 1




 Example:                              Example:
    Test the series for convergence    Test the series for convergence or
    or divergence.                     divergence.
                                                         
                                                             1
                                                         n1 / 3
               
                   1
               n3
              n 1
                                                        n 1
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS




                                   FINAL-081
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS

     THEOREM: (Integral Test)
     f (x) a continuous, positive, decreasing function on [1, inf)        f ( n )  an
         

         an Convergent
                                                   

        n 1
                                               1
                                                       f ( x ) dx Convergent

         

         an Divergent
                                                   

        n 1
                                               1
                                                       f ( x ) dx Dinvergent


                           REMARK:
  REMARK:
                             We should not infer from the Integral Test
  Integral Test just
                             that the sum of the series is equal to
  test if convergent
                             the value of the integral. In fact,
  or divergent. But if
  it is convergent              
                                    1 2                         1
  what is the sum??             n2  6
                               n 1
                                                          1      x 2
                                                                      dx  1
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
 REMARK:
                                                 
                                                    1 2                  1
                                                n2  6
  We should not infer from the Integral
  Test that the sum of the series is equal
                                               n 1
                                                                   
                                                                   1       x 2
                                                                               dx  1
  to the value of the integral. In fact,


 ESTIMATING THE SUM OF A SERIES
     

    a
    n 1
             n    a1  a2    an  an 1  an  2    
                        
                                            
                       n th partialsum
     

    a
    n 1
             n    a1  a2    an  s n
                        
                           
                       n th partialsum

 Example:
 Estimate the sum
    
        1
    n3
   n 1                         How accurate is this estimation?
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
 REMARK:
                                                  
                                                      1 2                  1
                                                  n2  6
  We should not infer from the Integral
  Test that the sum of the series is equal
                                                 n 1
                                                                     
                                                                     1       x 2
                                                                                 dx  1
  to the value of the integral. In fact,


 ESTIMATING THE SUM OF A SERIES

    

   a
   n 1
           n    a1  a2    an  an 1  an  2    
                      
                                          
                              n th partialsum               rem inder Rn

                 
    Rn         a
               i  n 1
                          i     an 1  an  2    
                                             
                                           rem inder Rn

    Rn  s  sn
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
    ESTIMATING THE SUM OF A SERIES
                                                        Rn  s  sn
    Rn     a
           i  n 1
                      i    an 1  an  2    
                                        
                                  rem inder Rn


  REMAINDER ESTIMATE FOR THE INTEGRAL TEST
                                                    
                     n 1
                             f ( x)dx  Rn   f ( x)dx
                                                     n
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
 REMAINDER ESTIMATE FOR THE INTEGRAL TEST                                          
                                                 sn         f ( x)dx  s  sn   f ( x)dx
n 1
        f ( x)dx  Rn   f ( x)dx
                        n
                                     Rn  s  sn        n 1                       n
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS




                                  TERM-102

				
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posted:9/26/2012
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