Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
f (x ) a continuous, positive, decreasing function on [1, inf)

 an Convergent
n 1

 an Divergent
n 1




1
1
f (n)  an
f ( x)dx Convergent
f ( x)dx Dinvergent
Example:
Example:
Test the series for
convergence or divergence.
Test the series for
convergence or divergence.

1

2
n
1
n 1

ln n

n 1 n
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
f (x ) a continuous, positive, decreasing function on [1, inf)

 an Convergent
n 1

 an Divergent
n 1




1
1
f ( x)dx Convergent
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

2
n  4 ( n  3)


4
f (n)  an
dx
(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)

 an Convergent
n 1
 an Divergent
n 1
1
REMARK:
f ( x)dx Convergent
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

2
n  4 ( n  3)


1




f (n)  an


4
dx
(x  3) 2
Also, it is not necessary that f(x)
be always decreasing. What is
important is that f(x) be ultimately
decreasing, that is, decreasing for
larger than some number N.
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
THEOREM: (Integral Test)
f (x ) a continuous, positive, decreasing function on [1, inf)

 an Convergent
n 1

 an Divergent
n 1




1
1
f (n)  an
f ( x)dx Convergent
f ( x)dx Dinvergent
Special Series:

Example:
For what values of p is the series
convergent?

1

p
n 1 n
 ar
1) Geometric Series
n 1
n 1

2) Harmonic Series

n 1
3) Telescoping Series
4) p-series

1

p
n 1 n
1
n

 (b b
n 1
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)

 an Convergent
n 1

 an Divergent
n 1
Example:




1
1
f (n)  an
f ( x)dx Convergent
f ( x)dx Dinvergent
P Series:
For what values of p is the series
convergent?

1

p
n 1 n
convg
1


p
n 1 n
 divg

p 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

p
n
n 1
Example:
convg
1


p
n 1 n
 divg

p 1
p 1
Example:
Test the series for convergence
or divergence.

1

3
n 1 n
Test the series for convergence or
divergence.

1

1/ 3
n 1 n
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)

 an Convergent
n 1
 an Divergent
n 1
Integral Test just
test if convergent
or divergent. But if
it is convergent
what is the sum??


1

REMARK:


1
f (n)  an
f ( x)dx Convergent
f ( x)dx Dinvergent
REMARK:
We should not infer from the Integral Test
that the sum of the series is equal to
the value of the integral. In fact,
1 2


2
6
n 1 n



1
1
dx  1
2
x
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
REMARK:
We should not infer from the Integral
Test that the sum of the series is equal
to the value of the integral. In fact,
1 2


2
6
n 1 n



1
1
dx  1
2
x
ESTIMATING THE SUM OF A SERIES

a
n 1
n
n th partial sum

a
n 1
 a1  a2    an  an 1  an  2    
 


n
 a1  a2    an  sn

n th partial sum
Example:
Estimate the sum

1

3
n 1 n
How accurate is this estimation?
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
REMARK:
We should not infer from the Integral
Test that the sum of the series is equal
to the value of the integral. In fact,
1 2


2
6
n 1 n



1
1
dx  1
2
x
ESTIMATING THE SUM OF A SERIES

a
n 1
n
Rn 
 a1  a2    an  an 1  an  2    
 


n th partial sum

a
i  n 1
i
Rn  s  sn
reminder Rn
 an 1  an  2    



reminder Rn
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
ESTIMATING THE SUM OF A SERIES
Rn 

a
i
i  n 1
Rn  s  sn
 an 1  an  2    



reminder 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


n 1

f ( x)dx  Rn   f ( x)dx
n
Rn  s  sn
sn  

n 1

f ( x)dx  s  sn   f ( x)dx
n
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
Sec 11.3: THE INTEGRAL TEST AND ESTIMATES OF SUMS
TERM-102