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### Transcript of Calculus I 5

• . 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

}

• 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.

• 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.

• 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

• 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

• 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

.