# Sec KFUPM by alicejenny

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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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