The Integral Test and p-series
Section 8.3 AP Calc
Thm 8.10 Integral Test
If f is positive, continuous, and decreasing
for x≥1 and an  f (n), then

a
n 1
n
and


1
f ( x)dx
either both converge or both diverge.
Use the Integral Test to determine the
convergence or divergence of the series:

3
1) 
n 1 2n  4

5
2)  2
n 1 n  1
Note: Just because an improper integral
converges to a value k, does not mean
that the series also converges to the same
value.

1
1
1
1
1
p-series:  p  p  p  p  p  ...
1
2
3
4
n 1 n
where p>0 is constant

1
1 1
Harmonic Series:   1    ...
2 3
n 1 n
(p-series with p=1)

1

n 1 an  b
Thm 8.11 Convergence of p-series

1
1
1
1
1
The p-series  p  p  p  p  p  ...
1
2
3
4
n 1 n
1) Converges if p>1
2) Diverges if 0<p≤1
Determine the Convergence/Divergence:

1
3)  8
n
n 1
1
1
1
 ...
5) 1  3  3  3
4
9
16

1
4)  2
n 1 n

5
6) 
n 1 n
Determine the Convergence or Divergence
of the series:

A)
n
n2

1
n 1
2

C)  (0.03)
n 0
B) 
n 1

n
1
n
0.03
 2 2
D)   3  5 
n 
n 1  n
Determine the Convergence or Divergence
of the series:

 1
E)  1  
n
n 1 
n

1
F) 
4
n 2 n(ln n)