Calculus I 5
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Transcript of Calculus I 5
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. 1 : a > 0, na
n 1
. 2 : nn
n 1
. 3 : xn > 0, xn n x
kxn
nkx
. 4 : P (n) = a0np + a1np1 + + ap1n+ ap,Q(n) = b0n
q + b1nq1 + + aq1n+ aq
p = q P (n)
Q(n)n
a0b0
p < q P (n)
Q(n)n 0
-
pi {yk}kN = {xnk}kN pi pi , n1 < n2 < n3 < pi {xn}nN
(x1, x2, x3, x4, x5, x6, x7, x8, x9, . . .
) (6x1 x2, x3, 6x4 6x5 x6, x7, 6x8 x9, . . .
) (xn1 , xn2 , xn3 , xn4 , xn5 , . . .
) (y1, y2, y3, y4, y5, . . .
) : k nk
-
pi.1:
xn x
pi
yk = xnk x
: { nn 1} {
n2n2 1
} Bolzano Weierstrass
( ) pi A
pi. 2:
( ) pi
-
pi {yk}kN = {xnk}kN pi pi , n1 < n2 < n3 < pi {xn}nN
(x1, x2, x3, x4, x5, x6, x7, x8, x9, . . .
) (6x1 x2, x3, 6x4 6x5 x6, x7, 6x8 x9, . . .
) (xn1 , xn2 , xn3 , xn4 , xn5 , . . .
) (y1, y2, y3, y4, y5, . . .
) : k nk
k N m N : {k m} {nk m}m N k N : {m nk}
k m nk m
-
pi.1:
xn x
pi
yk = xnk x
: { nn 1} {
n2n2 1
}
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pi.1:
xn x
pi
yk = xnk x
: { nn 1} {
n2n2 1
} pi: {
xn n
x}{ > 0, N() > 0 :k > N() |xk x| <
}pi nk k > N() |xnk x| <
{ > 0, N() > 0 :nk > N() |xnk x| <
}
-
BolzanoWeierstrass
( ) pi A
pi. 2 ( ) -
pi-
pi:I pi pi pi pi I pi pi pi x
n N, xkn : x1
n< xkn < x+
1
n
xkn n
x.
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xn xn xn+1xn xn xn+1xn xn
. 1
xn xn n sup{xk}kN
pi: xn xn+1 K. pi pipi, pi pi ` n ` xn = L R, pipi
L = limk
xk = max{x1, x2, . . . , xm} pi . R 3M = sup{xk}kN(pi R !) pi sup
> 0, k N : m < xk mxn n > k xk xn m < m+
pi > 0, k = N() : n > N() m < xn < m+
xn n
m
. 1
xn xn n inf{xm}mN
. 2xn pilimk
xnk = x xn xn n x
pi.
pi xnk (pi ) ,pi x = sup
kN{xnk}. x pi supremum
pi xnk (. . 1). A = {xnk , k N} B = {xn, n N}, z A z B pi z sup B x = sup A sup B. sup A < sup B ` : x` > x pi pi m xnm x` > x pi pi. pisup A = sup B.
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xn xn xn+1
xn xn xn+1
xn . 1: xn
xn n sup{xk}kN
. 1: xn xn
n inf{xm}mN. 2: xn -
pi limk
xnk = x xn xn
n x
-
I . xn =(1 +
1
n
)n
xn =
(1 +
1
n
)n=
n`=0
(n
`
)1
n`= 1 +
n
n+n(n 1)2!n2
+n(n 1)(n 2)
3!n3+ + 1
nn
=n`=0
n(n 1)(n 2) (n `+ 1)`!n`
=n`=0
1
`!
(1 1
n
)(1 2
n
)(1 3
n
) (1 ` 1
n
)I xn+1 > xn
xn+1 =
n+1`=0
1
`!
(1 1
n+ 1
)(1 2
n+ 1
)(1 3
n+ 1
) (1 ` 1
n+ 1
)
=n`=0
1
`!
(1 1
n+ 1
)(1 2
n+ 1
)(1 3
n+ 1
) (1 ` 1
n+ 1
)+
+1
(n+ 1)n+1(1 k
n+ 1
)>
(1 k
n
) xn+1 > xn
I . yn =n`=0
1
`!
I yn+1 > yn
-
I xn ynI yn xn
1
`!=
1
2 3 4 ` 1
2`1
xn < yn = 1 + 1 +1
2+
1
3!+ + 1
n!=
= 1 +
(1 + 12 + +
1
2n1
) 0
-
I . xn =(1 +
1
n
)n, xn+1 > xn
I . yn =n`=0
1
`!, yn+1 > yn ,
I xn ynI yn xn I xn
n e yn n e e e
I n,k =k`=0
1
`!
(1 1
n
)(1 2
n
)(1 3
n
) (1 ` 1
n
)I n,k xn e n,k
n yk I yk e lim
kyk = e
e
I e = e ! limn
(1 +
1
n
)n=k=0
1
k!
-
I(1 1
n
)nn e
1
I(1 +
k
n
)nn e
k
I(1 +
k
` n
)nn e
k/`
I(1 +
n
)n n e
= sup {eq, q Q, q }, > 0
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CAUCHY
{xn Cauchy
}
> 0 N() :
n > m > N() |xn xm| <
{xn
n x}{xn Cauchy
}{xn Cauchy
}
{xnk
kx
}
{xn
n x}
{xn Cauchy
}{xn
n x}
pi{xn Cauchy
}{xn
n x}
-
pi
pi{xn Cauchy
}{xn
n x}
{xn
}{xn Cauchy
} , N > 0 : n > m > N |xn xm| >
pi.
xn = 1 +1
2+
1
3+ + 1
n
xn = 1 +1
22+
1
32+ + 1
n2
|xn+1 xn| < k|xn xn1| k < 1 ! xn xn Cauchy
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CAUCHY
{xn Cauchy
}
> 0 N() :
n > m > N() |xn xm| <
{xn
n x}{xn Cauchy
}pi.{
xn n x
}{ > 0 N() > 0 :m > N() |x xm| <
}
n > m |xn x| < |xn xm| |xn x|+ |x xm| < 2 =
> 0 N () = N(2
):
n > m > N () |xn xm| <
{xn Cauchy
}
-
{xn Cauchy
}
{xnk
kx
}
{xn
n x}
pi.
{xn Cauchy
}
> 0 N1() :
n > m > N1() |xn xm| <
{xnk
kx
}
> 0 N2() :
k > N2() |xnk x| <
= 2 N () = max
{N1(), N2()
} n > nk > N ()
|xn x| |xn xnk |+ |xnk x| < 2 =
{ > 0 N () > 0 :n > N () |x xn| <
}{xn
n x}
{xn Cauchy
}{xn
n x}
pi.{xn Cauchy
}
= 1 n0 > N(1) : n > n0
|xn| |xn xn0 |+ |xn0 | < 1 + |xn0 | xn
Bolzano pi pi
xn
.
-
{xn Cauchy
}{xn
n x}
{xn
}{xn Cauchy
} , N > 0 : n > m > N |xn xm| >
pi.
xn = 1 +1
2+
1
3+ + 1
n
pi.
x2n xn = 1n+ 1
+1
n+ 2+ + 1
n+ n>
n
n+ n=
1
2
xn = 1 +1
22+
1
32+ + 1
n2
pi.
xn xm = 1(m+ 1)2
+1
(m+ 2)2+ + 1
n2
1m(m+1) + 1(m+1)(m+2) + + 1n(n1) == 1m 1m+1 + 1m+1 1m+2 + + 1n1 1n= 1m upslope1m+1 +upslope1m+1 upslope1m+2 + + 6 1n1 1n= 1m 1n < 1m
> 0 N() = 1
:
n > m > N() |xn xm| < 1m<
-
Cauchy
I R pi I Q pi pi. x1 = 2, xn+1 =
12
(xn +
2xn
)xn Q xn
n2 6 Q
Q pi pi Cauchy pi/pi R.R= Cauchy- pi Q
-
Constructive Axiomatic
PeanomN
pi Z
pipiQ
Cauchy- piR
, , mR
1-hereditarymN
pi Z
pipiQ
-
. (Stolz)anbnn `
bk > 0 nk=1
bk n
nk=1
ak
nk=1
bk
n `
pi xn n x
nk=1
xn
nn x
1 +2 + 33 + nnn
n n
k N 1 + 2k + 3k + nknk+1
n
1
k + 1
xn+1xn n ` n|xn| n `
pi.. limn
nn5 3n3 + 8 = 1
-
(Stolz):
anbnn `
bk > 0 nk=1
bk n
nk=1
ak
nk=1
bk
n `
pi:
anbnn `
{ > 0 N1() > 0 :k > m > N1() ` < ak
bk< `+
}
(` ) bk < ak < (`+ ) bk
(` )n
k=m+1
bk R
pi
xn n +
1xnn 0
{xn
n }R > 0, N(R) > 0 :
n > N(R) xn < R
pi
-
1 xn xn Cauchy
2 xn n 0 {xn}nN
C > 0 : n |xn| < C
3 |a| < 1 an n 0
4 () xn 6= 0, |xn+1xn | n k < 1 xn n 0
5 xn n x pi xnk x
6 (Bolzano-Weierstrass): ( ) pi
7 xn xn n sup{xk}kN
8 xn pi lim
kxnk = x xn
xn n x
9 limn
(1 +
x
n
)n= ex =
`=0
x`
`!
10 (Stolz):
anbnn `
bk > 0 nk=1
bk n
nk=1
ak
nk=1
bk
n `
11
xn+1xn n ` n|xn| n `
-
Sn =nk=1
ak ,
Sn n S pi S
k=1
ak SN Cauchy
> 0 N() : n > m > N() |Sn Sm| = nk=m+1 ak
<
pi n=1
an an n 0
: an
n 0 nan
pi. an =1
nn 0
n=1
1n =
-
pi pi k=1
|ak|
pi pi
(Comparison test)
|bn| an,n=1
an n=1
bn
0 an bn,n=1
an = pi n=1
bn = pi
-
(Ratio test)
n=1
an
1 pi limn
an+1an < 1
2 pi limn
an+1an > 1
3 pi pi limn
an+1an = 1
(Root test)
n=1
an
1 pi limn
n|an| < 1
2 pi limn
n|an| > 1
3 pi pi limn
n|an| = 1
-
an bn . limn
anbn
= ` 6= 0
n=1
bn pi n=1
an pi
an bn . limn
anbn
= 0
n=1
bn pi n=1
an pi
-
pi (Condensation test)
0 < an+1 < an :{ n=1
an pi}{ k=1
2ka2k pi}
:
p > 1 n=1
1np 1 n=2
1n(ln n)p
-
(1, 2, . . . , k, . . .) (1, 2, . . . , k, . . .)
k nk N : max {1, 2, . . . k} < nk
bm = am
nan pi
nbn pi .
-
a11 a12 a13 a14 upslope upslope upslope
a21 a22 a23 a24 upslope upslope upslope
a31 a32 a33 a34 upslope upslope upslope
a41 a42 a43 a44 upslope upslope upslope
m,n=0amn = ` pi
m=0
( n=0
amn
)= ` pi
-
Sn =nk=1
ak ,
Sn n S pi S
k=1
ak SN Cauchy
> 0 N() : n > m > N() |Sn Sm| = nk=m+1 ak
<
pi n=1
an an n 0
pi.
pi Cauchy
> 0, N() : n > N() |Sn Sn1| = |an| <
pi an n 0
: an
n 0 nan
pi. an =1
nn 0
n=1
1n =
-
pi pi k=1
|ak|
pi pi
pi.
{ pi }{ > 0, N() : n > m > N()
nk=m+1
|ak| < }
n
k=m+1
ak
n
k=m+1
|ak| < { > 0, N() : n > m > N()
n
k=m+1
ak
< }
-
(Comparison test)
|bn| an,n=1
an n=1
bn
pi.
n=1
an
> 0, N() :
n > m > N() nk=m+1 ak
<
nk=m+1
|bk|
nk=m+1
ak
> 0, N() :n > m > N()
nk=m+1
|bk| <
n=1
bn
pi
0 an bn,n=1
an = pi n=1
bn = pi
pi.
n=1
an = {R > 0, N(R) > 0 : n > N(R)
nk=1
ak > R
}nk=1
bk nk=1
ak > R n=1
bn =
-
(Ratio test)
n=1
an
(i) pi limn
an+1an < 1
(ii) pi limn
an+1an > 1
(iii) pi pi limn
an+1an = 1
pi (i) limn
an+1an
= c < 1 = 1c2 , n0 :n > n0
an+1 < (c + 1 c2
) |an|
|an| 1 = f12 , n0 :n > n0
an+1 > (f f 12
) |an|
|an| >(
f + 1
2
)n an0 (f+12
)n0pi f+1
2> 1 an
nan
(Root test)
n=1
an
(i) pi limn
n|an| < 1
(ii) pi limn
n|an| > 1
(iii) pi pi limn
n|an| = 1
-
pi (Condensation test)
0 < an+1 < an :{ n=1
an
}pi
{ k=1
2ka2k pi}
pi.
S2k+11 = a1 + (a2 + a3) 2
+(a4 + a5 + a6 + a7) 4
+
+(a8 + a9 + a10 + a11 + a12 + a13 + a14 + a15) 8
+ ++(a2k + a2k+1 + a10 + a2k+2 + + a2k+11
) 2k
a1 + 2a2 + 4a4 + 8a8 + + 2ka2k = Tk Tk pi S2k+11 Sn Sn .
Tk2
=a12+ a2 + 2a4 + 4a8 + + 2k1a2k
a1 + a2 + (a3 + a4) + (a5 + a6 + a7 + a8)++ ++(a2k1+1 + a2k1+2 + + a2k
)= S2k
Sn pi S2k Tk .
pin=2
1
n ln n
n=2
1
n(ln n)2
n=1
1
nr r > 1,
r 1
-
(Alternating SeriesTest)
0 < an+1 < an an n 0 :
{ n=1
(1)n+1an pi}
pi.
Sn =nk=1
(1)k+1akSm+2 Sm = (1)m+3am+2 (1)m+2am+1 = (1)m+2(am+1 am+2) |Sm+2 Sm| = am+1 am+2 0|Sm+4 Sm+2| = am+3 am+4|Sm+6 Sm+4| = am+5 am+6. . . . . . . . .|Sm+2p Sm+2(p1)| = am+2p1 am+2p|Sm+2p Sm| |Sm+2p Sm+2p2|+ |Sm+2p2 Sm+2p4|+ + |Sm+2 Sm||Sm+2p Sm| am+2p1 am+2p + am+2p3 am+2p2 + + am+3 am+4 + am+1 am+2|Sm+2p Sm| am+1 am+2p < am+1|Sm+2p+1 Sm| |Sm+2p+1 Sm+2p|+ |Sm+2p Sm| am+2p+1 + am+1 am+2p < am+1n > m |Sn Sm| < am+1limn an = 0
> 0 N() : m > N() am+1 < am <
|Sn Sm| < Sn Cauchy
pin
(1)nn
n
(1)nln n
n(1)n sin pi
n
-
nan pi
nbn pi .
pi.
:
(1, 2, . . . , k, . . .) (1, 2, . . . , k, . . .)
k nk N : max {1, 2, . . . k} < nk :
S` = |a1|+ |a2|+ + |a`| , Tk = |b1|+ |b2|+ + |bk|
pi bm = am pi Tk Snk . pi Snk Cauchy Tk Cauchy, . pi pi , pi .
Tk Snk limk
Tk limk
Snk n
bn n
an
an bn, pin
an n
bn
.