sequences

Question 1

sequence definition (general):

Answer 1

a list of real numbers, indexed by natural numbers
a1, a2, etc.

Question 2

sequence definition (specific):

Answer 2

a sequence is a function on the natural numbers to the real numbers
we write an(n in the natural numbers) for a sequence that maps n to an
the real number an is called the nth term

Question 3

infinite sequence:

Answer 3

not that the value set is infinite, but that the sequence has no end, nothing to make it stop, so 1,-1,1,-1… is infinite

Question 4

general term:

Answer 4

take 1,-1,1,-1… as an example
(-1^(n+1)))(n in the natural numbers) is the sequence
-1^(n+1) is the general term

Question 5

constant sequence:

Answer 5

a sequence that just repeats the same number r, r, r, r,…

Question 6

subsequence:

Answer 6

a sequence of the form (ank)(k in the naturals) where n1<n2<n3...>k for all k
basically just delete some members of the original sequence and voila, it's just which ones</n3...>

Question 7

is (a(2n)) a subsequence of an:

Answer 7

yes, with 2k=nk (replace ns with ks basically (I think))

Question 8

absolute value:

Answer 8

|a| is defined as a if a>=0, -1 if not
|a|<=b iff a<=b and -a<=b
|axb|=|a|x|b|
the triangle inequality holds
||a|-|b||<=|a-b|

Question 9

convergent sequence definition:

Answer 9

(an)n converges to a real number r if for every real number ε>0, there is some natural N such that for all n>=N, we have |an-r|<ε
if this is the case, we write an->r as n->∞
essentially ε is standing for anything more than 0, so |an-r|is close to 0, closer as n gets bigger

Question 10

divergent:

Answer 10

a sequence that uh. isn’t convergent to anything

Question 11

how to prove a sequence is convergent to a given limit:

Answer 11

write out |an-r|<ε with the given info
mess around to get smth like …ε…<n
then pick a natural number for N and a value for ε based on that to use
then I think you’re done? idk test some questions later

Question 12

limits are:

Answer 12

unique (no sequence has more than 1 limit) - take ε=(s-r)/2, where r and s are the ‘2’ limits, take the definitions of the sequence converging to each limit, combine for a contradiction

Question 13

finite modification theorem:

Answer 13

if a sequence (an)n converges to r, and (bn)n=(an)n for all but finitely many n, then (bn)n converges to r

Question 14

convergence and subsequences:

Answer 14

every subsequence of a convergent sequence converges to the limit of the convergent sequence

Question 15

scalar multiplication rule:

Answer 15

lim(c(an))=c(lim(an))

Question 16

sum rule:

Answer 16

lim(an)+lim(bn)=lim(an+bn)

Question 17

bounded sequences:

Answer 17

a sequence is bounded if there is a real number such that every element in the sequence is more/less/than equal to that real number
more specifically, it’s bounded from above or below, or just bounded if both

Question 18

bounds and convergence:

Answer 18

every convergent sequence is bounded
when n>=N, it’s bounded by r+-1
when n<N, it’s a finite sequence so bounded
union of 2 bounded sets is bounded so voila

Question 19

completeness axiom of the real numbers:

Answer 19

every nonempty subset of the real numbers that has an upper bound, has a least upper bound

Question 20

supremum:

Answer 20

the least upper bound
x=sup(S) <=> S<=x and for all r<x there is some s in S with r<s, S!<=r

Question 21

infimum:

Answer 21

inf(S)=-sup(-S)
greatest lower bound

Question 22

strictly increasing:

Answer 22

a sequence is strictly increasing if a1<a2<a3…

Question 23

strictly decreasing:

Answer 23

a sequence is strictly increasing if a1>a2>a3…

Question 24

strictly monotone:

Answer 24

a sequence is strictly monotone if it’s strictly increasing or decreasing

Question 25

increasing:

Answer 25

a sequence is increasing if a1<=a2<=a3…

Question 26

decreasing:

Answer 26

a sequence is decreasing if a1>=a2>=a3…

Question 27

monotone:

Answer 27

a sequence is monotone if it’s increasing or decreasing

Question 28

monotone convergence theorem:

Answer 28

every monotone and bounded sequence has a limit
assume (an)n is increasing, by assumption it has an upper bound, therefore it has a supremum which is the limit
if it’s actually decreasing, take (-an)n which is increasing and use scalar multiplication for the limit

Question 29

null sequence:

Answer 29

a sequence that converges to 0

Question 30

properties of a null sequence:

Answer 30

(an)n is a null sequence for these
if c is a real number, (can)n is a null sequence
if (bn)n is also a null sequence, (an+bn)n is a null sequence
if (bn)n is a sequence with |bn|<=|an|for all but finitely many n, (bn)n is a null sequence
if (bn)n is a bounded sequence, (an*bn)n is a null sequence
if an>=0 for all n, and p>0 is a real number, (an^p)n is a null sequence

Question 31

standard list of null sequences:

Answer 31

(1/n)n
(1/(n^p))n, p>0
(1/(c^n))n, |c|>1
((n^p)/(c^n))n, p and c are real numbers, |c|>1
((c^n)/n!)n, all real c

Question 32

the sandwich rule for sequences:

Answer 32

let (an)n, (bn)n, and (cn)n be sequences with these properties:
(an)n and (cn)n converge to the same r
(an)n<=(bn)n<=(cn)n
then (bn)n converges to r also

Question 33

limits and order:

Answer 33

if an<=bn for all but finitely many n, then lim(an)<=lim(bn)
if lim(an)<lim(bn), then for all but finitely many n, an<bn

Question 34

what functions can be moved within/without limits easily:

Answer 34

summation, multiplication, division, modulus, roots