数学分析高等数学微积分英语课件上海交通大学chapter11b

th1e)n
(3) sin p
n 1
n
bn 1/ n1/2
lni man /bn 2
(2) diverge. take
then
bn 1/ n
lni man /bn
(3) converge for p>1 and diverge for
take
then
lni man /bn p
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 )
satisfien s 1 (i)
for all n (ii)
Then the alternatibnng1serbiens is convergentln.im bn 0
divergence of a n .
Example
Ex. Determine whether the following series converges.

Sol.
(1) (1)
2n2 3n (2) ndi1ver5ge. nch5 oose
n1
ln2
1 (n
The n-th term2of a3 n alt4ernating nse1riesnis of the form
where
is aa n po s( it iv1 e)n n 1 ub mn beo rr . a n ( 1 )n b n
bn
The alternating series test
positive terms. Suppose
lim an c.
Then
b n n
(i) when c is a finite number and c>0, then either both series
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
p 1
bn 1/ np
Question
Ex. Determine whether the series converges or diverges.
ln 1 a n (a 0)
n 1
Sol.
an
ln1
a n
ln1lna
e n
1 nlna
divergefor 0ae
convergefor ae
Alternating series
An alternating series is a series whose terms are alternatively positive and negative. For example,
1111


(1)n1
lim an1 L 1,
a n n

bn
an bn
an
divergent.
1
n1 2 n 1
Ex.Determin1 ewhet1herconverges.

2n 1 2n


The limit comparison test
Theorem Suppose that a n and b n are series with
if
it
is
convergent but naont absolutely convergent.
Theorem. If a series is absolutely convergent, then it is
convergent.
Example
Ex. Determine whether the following series is convergent.
Thecomparisontests
TheoremSupposethatandareserieswithpositiveterms,then

an bn
(i)Ifisconvergentandforalln,thenisalsoconvergent.

bn
an bn
an
(ii)Ifisdivergentandforalln,thenisalso
Sol. (1) converge (2) converge
(2 )( 1 )n 1n2
n 1 n3 1
Question.
( 1) n 1 n
n1 4 n 1
Absolute convergence
A series
is called absolutely convergent if the series of
absolute values a n
is convergent.
| an |
For example, the series
( 1)isn a1 bsolutely convergent

while the A series
alternisatcinalglehdarcmonodnniitc1iosennra3iel/ l2syicsonnovt.ergent

(1)
sinn
n2
n1

(2)
(1)n
n1ln(1n)
Sol. (1) absolutely convergent
(2) conditionally convergent
The ratio test
The ratio test
(1) If (2) If (3) If
then
Ex. The alternating harmonic series is convergent.
( 1) n 1
n1 n
Example
Ex. Determine whether the following series converges.
(1 )n 1( n 1 )n 1 ( 0 )