real sequence
a function X: N ->R
to every n in N we assign a real number X(n) = Xn in R
we write (Xn) where n ∈ N
constant sequence
Xn = c for all n in N
convergent sequence
said to converge to a limit x in R if for every ε >0 there exists a n0 in N such that
|xn - x| <ε>= n0
we write lim n-> inf Xn = x
if no limit is divergent</ε>
eg prove Xn = 6 has a limit
chose x = 6 as a potential limit
for every ε >0 we have to find a good n0 in N
|Xn - 6| < ε for all n>= n0 where ε >0
look at |Xn - 6 | = 0 < ε (as Xn = 6)
so every n0 works
eg find limit of An = 1/n
chose a = 0
take ε >0
we need |1/n - 0| = 1/n < ε for all n >= n0
chose n0 > 1/ ε (by archimedes)
for all n >= n0
1/n <= 1/n0 < ε
eg bn = (-1)^n what would the limit be?
if x ∈ R what if x>= 0?
then |x - (-1)^2n+1 | = x+1 >= 1
so cant be < ε as ε <1
so cant get |x - bn| <ε
what is x<0?
|(-1)^2n -x| = 1-x >= 1 so cant work either
so no limit - divergent
uniqueness of a limit
a convergent sequence Xn has precisely one limit
bounded above/bellow/ bounded
let Xn be a sequence and X = { Xn ∈ R | n ∈N}
the sequence is bounded above is X is bounded above
bounded below if X is bounded below
and bounded if X is bounded
convergence and bounded
every convergent sequence Xn is bounded
contrapositive:
if Xn is not bounded then is divergent
squeezing theorem
let Xn and Yn be sequences
if |Xn| <= Yn for all n in N
and Yn -> 0 as n -> ∞
then Xn -> 0 as n -> ∞
COLT
calculus of limits theorem
Xn and Yn are convergent sequences with limit X and Y
* aXn + bYn -> aX + bY
* Xn.Yn -> X.Y
* Xn/Yn -> X/Y
all as n-> ∞
limit given a sequence within a range
XN is a sequence ∈ [a,b] for all n
if coverges them lim Xn ∈ [a,b] for n -> ∞
continuity of root
let Xn be convergent with Xn >= 0 then √Xn is also convergent
lim √Xn = √lim Xn (from n->∞)
limit and supremum
if Xn is a sequence monotonically increasing Xn <= Xm for all n<=m
if Xn is bounded then its convergent
lim Xn as n->∞ is sup{ Xn ∈R | n ∈N}
also works if monotonically decreasing and bounded (inf)
proof lim Xn is sup(X)
X = {Xn} is bounded
so sup(X) exists
let ε >0 we need |Xn -X| < ε for all n>= n0
x-ε cant be an upper bound so
x-ε < Xn0 <= Xn <= X < x +ε
-ε <Xn - X < ε
|Xn - X| <ε
exponential function
limit of (1+x/n)^n as n-> ∞
exp(x) >0 and exp(-x) = 1/exp(x)
what is exponential bounded by
if x is in R
then 1 + x <= exp(x)
if x<1
exp(x) <= 1/(1-x)
exp(x+y)
= exp(x) . exp(y)
exponential increasing or decreasing
stictly monotonically increasing
if x<y
exp(y-x) >= 1+ y-x > 1
so exp(y)/exp(x) >1
so exp(y) > exp(x)
when y > x
exponential continuous
let a >0 then there exists an x in R with exp(x) = a
logarithm function
log : (0,∞) -> R
is defined by log(x) = y where y is the unique real number with exp(y) = x
a^x in terms of log
exp(x.log(a))
log base a (x)
= log(x)/log(a)
a^log base a(x) = x
properties of powers and logs
a^(x+y) = a^x. a^y
(a^x)^y = a^xy
log(xy) = log(x) + log(y)
log(x^y) = ylog(x)
