Math 104 Final Notes

7/23/2019 Math 104 Final Notes

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Math 104 Final Review Sheet

Defnitions:

● subsequential limit = limit o a subsequence● Cauche c!ite!ion: o! Sn = sum"#$% !om 0 to n

& Sn is cauche& 'Sn ( Sm(1' ) e o! n*=m*+& 'sum"#$%m to n' ) e

● Continuit: o! eve! sequence in ,om conve!-in- to .0/ lim ".n% =

".0%● Continuit : cont2 at .0 3 o! eve! e*0 the!e is a ,*0 s2t2 . in

,om an, '. ( .0'), imlies '".% ( ".0%')e● unio!m continuit: o! eve! e*0 the!e is a ,*0 s2t2 i ./ in S an, '.('

), then '".% ( "%')e " on S%● lim.(*a s ".% = i o! subset S o R/ ,efne, on S/ o! all sequence in

S that conve!-es to a in S/ lim "sequence% = ● 5"% = su 5"/6%

7heo!ems:

● 102: boun,e, monotonic sequences conve!-e

8S 79R S75FF M8SS8+; 9R<

● 112:& subsequence o Sn conve!-in- to t 3 Sn within e o t is

infnite o! an e& Sn unboun,e, abovebelow =* has subsequence that

limits to >(oo● 112?: Sn conve!-es =* eve! subsequence has same limit● 1124: eve! sequence has a monotonic subsequence● 112@: AolBano eie!st!ass: eve! boun,e, sequence has a conve!-ent

subsequence● 112: an sequence has a subsequence that conve!-es to its limsu

an, one that conve!-es to its limin ● 112E:

& the set o subsequential limits is nonemt o! an

sequence& the su o this is is the limsu o Sn an, the in o this set

is limin o Sn& lim Sn e.ists 3 S has e.actl 1 elt which is limSn

● Fact: limsu = la!-est subsequential limit/ limin = smallest

subsequential limit

● 112: the limit o an sequence o!me, !om the fnite elements o theset o subsequential limits is also a subsequential limit "whethe! it is fnite o!

not%

● 121: limsu"Sn7n% = sGlimsu"7n% i limSn = s *0● 12: nonBe!o !eal Sn/ then limin'Sn>12Sn')=limin'Sn'H1n)=limsu'

Sn'H1n)=limsu'Sn>1Sn'● C12?: lim'Sn>1Sn' = I e.ists =* lim'Sn'H1n = I



● -eomet!ic se!ies: sum"aG!H$%0 to n = a G "1( !H"n>1%%"1 ( !%● sum"aG!H$% 0 to oo = a"1 ( !%● 1424: conve!-ent se!ies 3 satisfes cauche c!ite!ion

& C142@: sum"#n% conve!-es =* lim #n = 0● 142J: Coma!ison 7est: i #n *= 0

& sum"#n% conve!-es an, 'An' )= #n =* sum An conve!-es& sum"#n% = >oo an, An *= #n =* sum"An% = >oo& C142: absolute conve!-ence =* conve!-ence

● 142E: Root 7est: #n nonBe!o& conve!-es absolutel i limsu'#n>1#n' ) 1& ,ive!-es i limin'#n>1#n' *1& else no ino

● 142: Root 7est/ K = limsu'#n'H1n& sum"#n% conve!-es absolutel i K)1& ,ive!-es i K*1& K = 1 =* no ino

● 1@21: sum"1nH% conve!-es 3 *1● 8nte-!al test● 1@2?: alte!natin- se!ies test/ a1*a*222 an, liman = 0/ then

sum""(1%HnGan% conve!-es an, 's ( sn') an● 12?: cont at .0 in ,om =* '' an, $ a!e cont2 at .0● 124: >-/ -/ - a!e cont2 at .0 i an, - a!e cont2 at .0 "last

one onl i - L= 0%● 12@: cont2 at .0/ - cont at ".0% then -"".%% cont2 at .0● 1E21: cont2 an, boun,e, on a/bN then the!e a!e .0 an, 0 in

a/bN s2t2● 1E2: cont2 on 8 then o! an a/ b in 8 the!e is an . between a

an, b s2t2 ".% is between "a% an, "b%& C1E2?: cont2 on 8 =* "8% is an inte!val o! oint

● 1E24: cont/ st!ictl in on 8 =* inve!se is continuous/ st!ictl

inc!easin- on O "which is its ,omain/ ="8%%● 1E2@: - st!ictl in on O/ ;"O% = 8 =* - is continuous on O● 1E2J: 1(to(1 cont2 on 8 then is st!ictl inc o! ,ec● 12: continuit on a close, fnite inte!val =* unio!m continuit● 124: u(cont2 on S an, "Sn% cauche =* ""Sn%% cauche● 12@: u(cont on "a/b% 3 can be e.ten,e, to a cont2 unction on

a/bN● 12J: cont2 on 8/ ,iP on 8 wo en,oints "i the!e a!e an% an, Q

boun,e, =* u(cont2 on 8● 024: lim ,ist!ibutes ove! unction a,,ition/ multilication/

,ivision/ as lon- as both limits a!e fnite● 02@: lim -"".%% = -"lim ".%% i lim ".% e.ists an, is fnite



● 02J: lim ".% = I 3 o! eve! e*0 the!e is a ,*0 s2t2 . in S subset

o R/ a lim o a seq in S/ '. ( a' ) , =* '".% ( I' ) e& C02: still wo!$s i S is missin- a& C02E lim !om the !i-ht 3 same thin- but use 0 )

.) a > ,

● 0210: S missin- a/ limit e.ists i lim> an, lim( both e.ists an,a!e equal

● E2: ,iPe!entiable at a =* cont at a● E2?: c/ > -/ -/ - ,iP i /- ,iP ● E24: chain !ule/ ""-".%%%Q = Q"-".%%G-Q".% i - ,iP at . an, ,iP at

-".%● 21: Q"ma. o! min% = 0 i ,efne, on oen inte!val containin-

the oint an, ,iP at the oint● 2"Rolle%: cont on a/bN/ ,iP on "a/b%/ "a% = "b% then Q".% = 0

o! some . in "a/b%

● 2?"M7%: cont on a/bN/ ,iP on "a/b%/ then Q".% = "b% ( "a% "b(a% o! some . in "a/b%

& C24: Q".% = 0 an, ,iP on "a/b%/ then const on "a/b%& C@: Q = -Q then an, - ,iPe! b a constant i both

,iP/ on "a/b%● 2: Q *0 =* st!ictl inc/ / as lon- as ,iP on "a/b%● 2E: .1)./ both in "a/b%/ then the!e is an . between .1 an, .

s2t2 Q".% = c o! an c in "Q".1%/ Q".%% i is ,iP on "a/b%● 2: 1(to(1 on oene, 8/ O = "8%/ ,iP at .0 in 8 an, Q".0% L= 0/

(1 is ,iP at 0 = ".0%/ then "(1%Q"0% = 1Q".0%● ?021: / - const/ ,iP on "a/b%/ the!e is an . in "a/b% s2t2

& Q".%"-"b% ( -"a%% = -Q".%""b% ( "a%%● ?02: to al lQhosital/ nee,

& lim Q".%-Q".% e.istsis fnite an,& lim ".% = lim -".% = 0 o! lim '-".%' = >oo

● ?2?: I"/6%)=5"/T% o! an a!titions 6 an, T i boun,e,● ?24: I"% )= 5"% i boun,e,● is inte-!able on a/bN i o! eve! e*0 the!e is a 6 s2t2 5"/6% (

I"/6% ) e an, is boun,e,& inte-!able ,efne, as 5"% = I"%

● ?2J: is inte-!able i o! e*0 the!e is a , s2t2 mesh6 ) , =*

5"/6% ( I"/6% ) e o! an 6 an, boun,e,● Riemann inte-!able i the!e is an ! s2t2 o! eve! e*0 the!e is a

,*0 s2t2 'S ( !' ) e o! an Riemann sum s o w with a!tition 6 with

mesh ) ,● ?2: Riemann inte-!able 3 ,a!bou. inte-!able



● ?210: boun,e, Riemann inte-!able/ the limit o an sequence

o Riemann sums whose mesh conve!-es to Be!o conve!-es to the

inte-!al o ove! the inte!val● ??21: all monotonic on a/bN a!e inte-!able● ??2: cont2 on a/bN =* inte-!able

● ??2?: / - inte-!able on a/bN then c/ > - inte-!able● ??24: )= - then inte-!al o )= inte-!al o -● ??2@: inte-!able on a/bN then '' inte-!able an, 'inte-!al"%' )=

inte-!al ''● ??2J can stitch to-ethe! boun,s● ??2E: iecewise monotonic o! iecewise cont2 on a/bN =*

inte-!able

Facts

● U @24: in(S = (suS● U 112E: liminSn = ( limsu"(Sn%● U 12: limsu'Sn' = 0 3 limSn = 0● U 124: limsu "Sn > 7n% )= limsuSn > limsu7n● continuous!eal value, =* non(infnite● lim'An' = 0 3 limAn = 0● ma."/-% = ">-% > ' ( -'● unio!m cont2 =* cont2● !o,uct o two u(cont unctions is not u(cont "i2e2 . an, . =* .H%● 02:

& cont )=* lim ".n% = ".% i .n conve!-es to . an, ".% is

,efne,& limits a!e unique "o! unctions an, sequences%

● ,omQ is a S5AS7 o ,om/ even i it loo$s li$e itQs ,efne,● U124: u(cont on a boun,e, set/ then i boun,e, on that set● 5"(/6% = (I"/6%

St!ate-

● 7o !ove unction lim i,entities/ conve!t into sequence ,e an, use

e.istin- sequence theo!em

6ost(M:

Defnitions:

● 6ointwise Conve!-ence: lim n".% = ".% 3∀ ϵ>0∃ N s .t .∨fn( x )−f ( x)∨¿ϵ "two equivalent ,es%

● 5nio!m Conve!-ence:

∀ ϵ>0∃ N s .t . x∈S∧n> N =¿∨fn( x )−f ( x)∨¿ϵon intervalS

● 5nio!ml Cauche: ∀ ϵ>0∃ N s .t .∨fn( x )−fm( x)∨¿ϵ for all m, n> N

● Cauche C!ite!ion: ∀ ϵ>0∃ N s .t . m≥ n> N →∨∑k =m

n

gk ( x )∨¿ϵ



● 7alo! Se!ies: ∑ (f k (0) xk /k !)

● Met!ic sace: ,"./% ≥ 0/ ,"./% 3 . = / ,"./% = ,"/.%/ ,"./B% ≤

,"./% > ,"/B%● Aoun,e,: all . in S a!e )= A awa !om a oint ● Vene,: eve! oint has an oen ball a!oun, it which is in the set

● Close,: comlement is oen● Met!ic Sace: ,"./% ≥0 / ,"./% = 0 3 . = / ,"./% = ,"/.%/ ,"./B%

≤ ,"./% > ,"/B%● Connecte,: onl oen an, close, subsets a!e itsel an, emt● Closu!e: inte!section o all close, sets containin- #● 8nte!io!: oints that have an oen ball a!oun, them which is in the set● Aoun,a!: closu!e# W 8nte!io!#

7heo!ems: "al on an inte!val usuall%

● ?21: owe! se!ies conve!-e o! '.' ) R/ ,ive!-e o! '.' * R/ R = 1b/ b

= limsu'an'H"1n%● 42?: unio!m limit o cont2 unctions is cont2 "use this to show

continuit o comle. unction%● !ema$e 424: "n% conve!-es unio!ml to 3 limsuX'".% ( n".%':.

∈S }=0

● @24: unio!ml cauche =* ∃ f s . t . fn→ f uniformly cont ● @2@: the unio!m limit o a se!ies continuous unctions is continuous● @2J: unction se!ies satisfes CC =* conve!-es unio!ml

& Co! @2 "eie!st!auss M(test%: "M$% seq o non(ne- !eals/

sumM$ ) oo/ '-$".%' )= M$ o! all . in S then sum -$ conve!-es

unio!ml on S● J21: owe! se!ies conve!-e unio!ml on (R1/ R1N to a con2t unction

i R*0 an, 0)R1)R& Co! J2: sum"anG.Hn% conve!-es to cont2 unction on "(

R1/ R1%● Iemma J2?: sum"anG.H% has R =* sum"nGanG.Hn(1% an, sum "an

G.Hn>1 n>1% also have R● J24: unction sum has R*0 then inte-!al !om 0 to . is what ou

e.ect on '.')R● J2: unction sum has R *0/ then it is ,iP on "(R/R% an, ,e!ivative is

what ou e.ect o! .)R● J2J: owe! se!ies conve!-e at x=± R if x cont . there∧ R finite

● ?12? "7alo!Qs%: x−c ¿n/n !

∀ c ≠ c , x∈(a , b)∃ ybeteen canx s. t . Rn( x)=f n( y)¿

if f !efine! (infinite okay )∧a<c<b , f n

!efine! on(a , b)& ?124: limRn".% = 0 on "a/b% i ,efne, an, all ,e!s e.ist/

a!e boun,e, b some C● 2124 "Cauche(Schwa!tB%: x , y∈"

n then sum".YY%H ≤

sum".YH%G"sum"YH% summin- Y to n● 212: subsaces o MS is MS● 22J: emt set an, U a!e oen/ an, fnite inte!section an, an union

o oen sets is oen



● 22E: emt an, U a!e close,/ an inte!section an, fnite union o

close, sets is close,● 22: oen ball is oen an, close, ball is close,● 2210: "a/b%/ "a/oo%/ "(oo/b% a!e oen/ a/bN/ a/ooN/(oo/bN a!e close,● 221: S ,isconecte, 3 nonemt an,

∃u1 , u2o#ens.t.$ 1%$ 2% S=

⊘,$ 1% S ≠

⊘$ 2% S ≠

⊘S=($ 1% S)

∪

($ 2 %S )=S● 2214: S in R is connecte, 3 S is an inte!val o! a oint● 221: closu!e# is close, an, = # i # close,● 220: . in closu!e# 3 A"./,% inte!sects # o! all ,● 224: closu!e is close, an, inte!io! is oen● 22J: ,# = closu!e# inte!section closu!e"#c%

Facts

● 0H0 = 1● lim'an>1an' = b i it e.ists● limit o cont2 a!tial sums mi-ht not be cont2 "i2e2 1 ( '.'%Hn%● sin"n.%n →0

● -/ h inte-!able on a/bN an, - ) h/ then ∫a

b

g( x )x<∫a

b

h ( x)x

● - inte-!able on a/bN →∨∫a

b

g( x )x∨≤∫a

b

¿ g( x )∨x

● n =* unio!ml on a/bN then lim∫a

b

fn( x)!x=∫a

b

f ( x)!x

● ".% = sum"$"0%G.H$$L%/

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Math 104 Final Notes