9.3 Integral Test and P-Series
p-series Test

1
1 1
1
 p  p  p  

p
1 2
3
n 1 n
converges if
p  1 , diverges if p  1 .
We could show this with the integral test.

the harmonic series:

1 1 1 1 1
     

1 2 3 4
n 1 n
diverges.
(It is a p-series with p=1.)
It diverges very slowly, but it diverges.
Because the p-series is so easy to evaluate, we use it
to compare to other series.

Use a P-Series test to determine if the
series appears to converge
a) ¼ + 1/9 + 1/16+ 1/25+ …
_
_
_
b) 1 + 1/√4 + 1/√9 + 1/√25 + …
Use a P-Series test to determine if the
series appears to converge
∞
a) ¼ + 1/9 + 1/16+ 1/25+ … = Σ 1/n2
∞
n=2
Σ(1/n)2 Is a p series with p= 2 converges
n=2
_
_
_
∞
b) 1 + 1/√4 + 1/√9 + 1/√25 + … = Σ 1/(n) 1/2
n=1
Is a p series with p = ½ diverges
Remember that when we first studied integrals, we used a
summation of rectangles to approximate the area under a
curve:
3
2
This leads to:
1
0
1
If
2
3
4
The Integral Test
an  is a positive sequence and an  f  n  where
f  n  is a continuous, positive decreasing function, then:

a
n N
n
and


N
f  x  dx
both converge or both diverge.


Example 1:
Does
n
1
n
n 1


1
1
x x
dx
converge?
b
 lim  x
b  1

3
2
dx
 lim
b 
2 x
1 b

2
1
 2

 2  2
 lim 
b
 b

Since the integral converges,
the series must converge.
(but not necessarily to 2.)

Flow chart (so far)
Does a series converge?
Does the nth term approach zero → no diverges
If Yes or maybe
Is it a special series
Geometric series →yes converges if │r│ < 1 to a/(1- r)
P series
→yes converges if p >1
Telescoping series →yes converges to uncancelled terms
If No
Use a general test
The only general test that we have so far is the Integral test
The series converges if the corresponding integral converges
(only works for series that are positive, continuous and decreasing)
Use any method learned so far to determine
if a series converges or diverges
∞
∞
∞
Σ (1/n.95)
Σ (1/n2 - 1/(n+1)2)
Σ x/lnx
n=1
n=1
n=1
∞
∞
Σ 4n/(2n2+1)
Σ (n+1)/(n+5)
n=1
n=1
Use any method learned so far to determine
if a series converges or diverges
∞
Σ (1/n.95)
∞
Σ (1/n2 - 1/(n+1)2)
∞
Σ x/lnx
n=1
n=1
n=1
Diverges
P-series
Converges
Telescoping
Diverges
nth term test
∞
∞
Σ 4n/(2n2+1)
Σ (n+1)/(n+5)
Diverges
Integral test
Diverges
nth term test
n=1
n=1
p. 622 1-5 odd,17, 27-41odd, 87-97 odd