Sequence - math.berkeley.eduapaulin/1BSequences.pdf · Monotone Convergence Theorem au bounated and...
Transcript of Sequence - math.berkeley.eduapaulin/1BSequences.pdf · Monotone Convergence Theorem au bounated and...
Sequencelist of numbers
Sequence a ar as 94 with a definiteorder
Visualizations Alternate notation
I Eau or ann _I
12 as0 a a 2 a
a
Fundamental Examples 1 auction y tcxL
IN 1,2 3 3 tch
Imputantpeitiesy Eau is bounded above it there exists 14 such that
Man C M 7 u all n
4 Eau is bounded below it there exists N such that
an N 7 u all U Iau is bouncted if it is both bounded above and
below K
tK
Ly Eau is in it au Eau 7 u all n
S Eau is decreasing it an an 7 u all nI
12µsIf au is increasing decreasing we say it is monotone
We sayda is eventually bounded monotone
so
it an Nis For some N
F bounded monotone f n bounded monotone
Limitquencesau sequence an approaches L as a growsInformal positively without bound
measures how close toim an L we are
n s x Given E o there existsI ea.se N such that law Ll E
Limit tu all u N Ta number means an is in
Visualizations L E Lt En Lsu IL E 1
C E q La a L E
µan anti N
RemarksIn both cases
We have usual definitions of Limit DIVE
in an N Lim an an a u s
Lim 7cal L fol a Lim 7cal I Ic s j u a q
Limit of sequenceLimit of function
nExample an U t l an f u where
1 X1 L HospitalCt l
L 1im 7 ex Lim I Lim au I
i nx a x a
Important Theorems
If Lim an L and F is continuous at Ln s
Elen Lim 1 Can 7 Lim au 7 Ln n s
squeeze Theorem If Cerentally an E bn E caElen
Lim an Lim Cu L Lim bn Lun s a s
Monotone Convergence Theorem
au bounated and monotone Lim an L For someu
number LPicture
9 are aF.in
L M c upper bound9must be some limit point with LEM