数学分析 高等数学 微积分 英语课件 上海交通大学 chapter11b

111111 ln2. 23456
Multiplying this series by 1 / 2 , we get
111111 1ln2. 2 4 681012 2 or 01 01 01 0 11ln2 .
2468 2
Adding these two series, we obtain
11111 3
(2) If
lim an1 a n
n
L 1 or
lim
n
a n 1 an
then a n diverges.
n 1
(3) If lim a n 1 1, the ratio test is inconclusive: that is, no a n
n
conclusion can be drawn about the convergence of a n
an
1 n(lnn)(lnlnn)
1 ( 1 ) n ln n 2 n n !
1
( 1 ) ta n( 2 ) n
n
( 3 )
( 4 )
( n 2 ) !
( ln n ) ln n
Sol. (1) diverge (2) converge (3) diverge (4) converge
where b n is a positive number.
h
5
The alternating series test
Theorem If the alternating series
( 1 )n 1 b n b 1 b 2 b 3 b 4 b 5 b 6 (b n 0 )
n 1
satisfies (i) bn1 bn for all n
A power series may converge for some values of x and diverge for other values of x. So the sum of the series is a function s (x ) c 0 c 1 x c 2 x 2 c 3 x 3
h
18
Power series
For example, the power series
xn 1xx2x3
n0
converges to s ( x ) 1 when 1x1. 1 x
(3)
If
lim
n
n
|
an
|
1,
the
root
test
is
inconclusive.
h
12
Example
n
Ex. Test the convergence of the series
n 1
a
1 n
n
Sol.
lim n
n
an
lim n n n a 1
1 a
n
(a 0)
convergent for a 1; divergent for 0a1
a1an
n (11)n
(n)
n
h
13
Rearrangements
If we rearrange the order of the term in a finite sum, then of course the value of the sum remains unchanged. But this is not the case for an infinite series.
ae an 1an lni m an 0
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11
The root test
The root test
(1)
If
limn
n
|
an
|
L1,
then
a n is absolutely convergent.
n 1
(2)
If
limn
n
|
an
|
L1or
limn
n
|
an
|
then
n
1
a
n
diverges.
(1 )n 1( n 1 )n 1 ( 0 )
(2 )n 1( n 1 3 )n 1 1 n2
Sol. (1) converge (2) converge
Question. ( 1) n 1 n
n1 4 n 1
h
7
Absolute convergence
A series a n is called absolutely convergent if the series of absolute values | an | is convergent.
( 1) n 1
Fwohrileexathmepallet,erthneatsinergiehsanrm1 onni3c/
2 is series
absolutely is not.
convergent
A series a n is called conditionally convergent if it is convergent but not absolutely convergent.
Theorem. If a series is absolutely convergent, then it is convergent.
h
8
Example
Ex. Determine whether the following series is convergent.
(1)
sinn
n2
n1
converges or diverges.
n 1
(a 0)
Sol.
an
ln1
a n
ln1lna
e n
1 nlna
divergefor 0ae
convergefor ae
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4
Alternating series
An alternating series is a series whose terms are
same sum s .
However, any conditionally convergent series can be rearranged to give a different sum.
h
14
Example
Ex. Consider the alternating harmonic series
or a form containing factorial, then use comparison test.
an
n31 ~ 1 3n34n22 3n3/2
For an alternating series, use alternating series test.
an
(1)n
n3 n4 1
converge or both diverge.
(ii) when c=0, then the convergence of b n implies the convergence of a n . (iii) when c , then the divergence of b n implies the
h
16
Strategy for testing series
If n-th powers appear in the series, use root test.
an nen2
If an f (n), f decreasing and positive, use integral test.
(ii)
lim
n
bn
0
Then theቤተ መጻሕፍቲ ባይዱalternating series is convergent.
Ex. The alternating harmonic series ( 1) n 1
is convergent.
n1 n
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6
Example
Ex. Determine whether the following series converges.
(2) diverge. take bn 1/ n then lni man /bn
(3) converge for p>1 and diverge for p 1 take bn 1/ np
then lni man /bn p
h
3
Question
ln 1
Ex. Determine whether the series a n
h
17
Power series
A power series is a series of the form
cnxnc0c1xc2x2c3x3
n0
where x is a variable and c n are constants called coefficients
of series.
For each fixed x, the power series is a usual series. We can test for convergence or divergence.
1 ln2.
3 2 5 7h 4 2
15
Strategy for testing series
If
we
can
see
at
a
glance
that
lim
n
an
0
then
divergence
an
n1 10 2n1 2
If a series is similar to a p-series, such as an algebraic form,