10.3
p-Series and the Ratio Test
Objectives
Determine the convergence or divergence of
series with nonnegative terms.
 p-series.
 Comparison Test and Limit Comparison Test .
 The Ratio Test
1. series with nonnegative
terms
正項級數歛散性的判別法
2
Recall 有界單調數列必收
3
正項級數歛散性之通則

a
, Since an ≧ 0 , So, Sn ↗
Sn convergent
Sn bounded(above)
正項級數
n 1
n
Thm :

an ≧ 0 ,
a
n 1
n

<∞ ⟹
a
n 1
n
Conv.
4
Integral Test : f(n)=an
5
2. p-series
6
p-series
In this section you will study another common type of series
called a p-series.
Power function with
negative power
7

p=2
1
1 1 1 1 1
 2  2  2  2  2  ...

2
1 2 3 4 5
n 1 n
f (x) =1/x2 on
[1, ∞)

1 1 1 1 1
1
 2  2  2  2  ...   2
2
1 2 3 4 5
n 1 n
 1
1
  2 dx  2
2
1 x
1
8
p = 2 conv.

1
1 1 1 1
 2  2  2  2  ...  2

2
1 2 3 4
n 1 n
The exact sum of this series was found by the
mathematician Leonhard Euler (1707–1783) to be p2/6.
9
p = ½ div.
1
1
1
1
1




 ... 
1
2
3
4
5


1


n 1
1
n
(1/ x ) dx = ∞
Thus, the sum of the series must be infinite.
That is, the series is divergent.
10
p-series
You can determine the convergence or divergence of a pseries.
an 趨近0 的階小,速度太慢
an 趨近0 的階大,速度夠快
For p = 1, this series is the harmonic series
The harmonic series diverges.
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
1
Discuss p for series  p
n 1 n
p
lim
(1
/
n
)
＊ If p < 0, then n  
p
lim
(1
/
n
) 1
＊ If p = 0, then
n 
p
lim
(1
/
n
)0
 In either case, n  
– So, the given series diverges by the Test for
Divergence Only need to consider p>0
12
p-series , p > 0
con’d
If p > 0, then the function f(x) = 1/xp is clearly continuous,
positive, and decreasing on [1, ∞).


1
1
dx
p
x
It follows that the series
∑1/np converges if p > 1 and diverges if 0 < p ≤ 1.

1

p
n 1 n
div.
0
1
Conv.
p
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Example 2 – Determining Convergence or Divergence
Determine whether each p-series converges or diverges.
a.
b.
c.
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Example 2(a) – Solution
For the p-series
p = 0.9.
Because
you can conclude that the series diverges.
15
Example 2(b) – Solution
cont’d
For the p-series
p = 1, which means that the series is the harmonic series.
Because
you can conclude that the series diverges.
16
Example 2(c) – Solution
cont’d
For the p-series
p = 1.1.
Because p > 1, you can conclude that the series converges.
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3. Comparison Test and
Limit Comparison Test
18
Geomety series and p-series
Question:
19
Comparison Test (補充1)
欲知
已猜
去找來比較的
證明完成 猜測正確
20
Ex: determine conv. or div.
1
Show
that 
diverges.
EX 4:
3k  1
? EX 5:
21
(補充2)
欲知
去找來比較的
趨近0的階一樣大
趨近0的階an 比較大
趨近0的階bn 比較大
22
Examples
Cont,d
5.

6.
1

k
5
3
k 1
7.
3k 2  2k  1
 k 2 1
8.

2k  5
k 6  3k 3
?
23
3. The Ratio Test
24
絕收必收
25
The Ratio Test
The next test is more general: it can be applied to infinite
series that do not happen to be geometric series or p-series.
1
擬公比 r
div.
Conv.
0
=
1
失效
擬公比
26
Example
27
Example 3 – Using the Ratio Test
Determine the convergence or divergence of the infinite
series
Solution:
Using the Ratio Test with
you obtain
28
Example 3 – Solution
cont’d
Because this limit is less than 1, you can conclude that the
series converges.
Using a graphing utility, you can approximate the sum of
the series to be
29
The Ratio Test
擬公比=1 失效
When applying the Ratio Test, remember that when the
limit of
as
is 1, the test does not tell you
whether the series converges or diverges.
This type of result often occurs with series that converge or
diverge slowly.
Ex: The Ratio Test is inconclusive for any p-series.
倒數就是無窮小階
無窮大的"階”
nn
n!
階乘
很強
2n
指數
函數
n2
n (0    1) ln n
正的 Power fct
對數最弱
30
Examples for
擬公比=1 為何失效
Σ1/n2 convergent, we have:
an 1
an
1
n2
1
( n  1) 2



1
2
2
1
( n  1)
 1
2
1  
n
 n
as n  
Σ1/n divergent, we also have
an 1 / an  1 ,
Therefore, if lim
n 
the series Σan might converge or it might diverge.
31
Example 5 – Using the Ratio Test
Determine the convergence or divergence of the infinite
series
Solution:
Using the Ratio Test with
you obtain
32
Example 5 – Solution
cont’d
Because this limit is greater than 1, you can conclude that
the series diverges.
33
Summary of tests for series 1
看自己:
34
Summary of tests for series 2
跟別人比:
1.Integral test
2. Comparison Test
3. Limit Comparison Test
35
10.3 Homework
上完課會公告 Exercises 10.3 的題目
請交第 13, 18, 23, 27, 29, 31,
47, 49, 53, 55, 59, 61
共12 題
36
Test : determine the conv. or div.
5.

1
1
1
1
1




 ...  
1
2
3
4
5
n 1
37