(a,b)
a ≤ x ≤ b
{x ∈ R : a ≤ x ≤ b }
{x ∈ R : a < x < b }
a < x < b
(4,1)
What are sets with curved brackets named?
Open Intervals
Closed Intervals
What are sets with square brackets named?
State the triangle inequality
|a + b| ≤ |a| + |b|
a + b ≤ |a| + |b|.
|a + b| < |a| + |b|
|a + b| ≤ |a| - |b|.
Define what is meant by a sequence
Corresponds to a mapping (or ) from the natural numbers N to the real numbers R.
Ordered list
Numbers in a set
corresponds to a mapping (or ) from a number to another
corresponds to a mapping (or ) from the real numbers R to the real numbers N.
Corresponds to a mapping (or ) from the natural numbers N to the integers Z.
An increasing list of values mapped from the Natural numbers N to the integers Z
Define tends to infinity
A sequence (an) of real numbers tends to infinity if given any real number A > 0 there exists N ∈ N such that an>A for all n>N.
It gets bigger and bigger past a number
A sequence (an) of numbers tends to infinity if given any number A > 0 there exists N ∈ N such that an>A for all n>N.
A sequence (an) of real numbers goes to infinity if given any real number A > 0 there exists N ∈ N such that an>A for all A>N.
A sequence (an) of real numbers tends to infinity if given any real number A > 0 there exists N ∈ N such that an>A for some A>N.
A sequence (an) of real numbers tends to infinity if given any real number A > 0 there exists z ∈ Z such that an>A for all A>N.
Define Tends to infinity
∀A>0 ∃N∈N s.t. an>A ∀n>N
∃A>0 ∃N∈N s.t. an>A ∀n>N
∀A>0 ∀N∈N s.t. anN
∀A>0 ∃N∈N s.t. an>A ∀n<N
What is |x|^2 equal to ?
x^2
x
|x|
-x
What's another way to write √(x^2)
|x+1|
|xy| =
|x||y|
xy
|x+y|
|x|+|y|
Define Convergent sequence
A sequence (an) of real numbers converges to a real number ℓ if given any e > 0 there exists N ∈ N such that |an − ℓ| < e for all n > N
A sequence (an) of numbers converges to a real number ℓ if given any e > 0 there exists N ∈ N such that |an − ℓ| < e for all n > N
A sequence (an) of real numbers converges to a number ℓ if given any e > 0 there exists N ∈ N such that |an − ℓ| < e for all n > N
A sequence (an) of real numbers converges to a real number ℓ if given any e < 0 there exists N ∈ N such that |an − ℓ| < e for all n > N
A sequence (an) of real numbers converges to a real number ℓ if given any e > 0 there exists Z ∈ N such that |an − ℓ| < e for all n > N
A sequence (an) of real numbers converges to a real number ℓ if given any e > 0 there exists N ∈ N such that |an − ℓ| < e for some n > N
A sequence (an) of real numbers converges to a real number ℓ if given any e > 0 there exists N ∈ N such that |e − ℓ| < e for all n > N
(Converging series) If |an-l| = 1/n. What should you let N be greater than?
1/e
e
2e
2/e
Define bounded above
if there exists some M ∈ R such that an ≤ M for all n ∈ N
if there exists some M ∈ N such that an ≤ M for all n ∈ N
if there exists some M ∈ R such that an ≤ R for all n ∈ N
if there exists some M ∈ R such that an ≤ M for some n ∈ N
if there exists some M ∈ R such that an ≤ M for all R ∈ N
Define bounded below
there exists some M ∈ R such that an ≥ M for all n ∈ N.
there exists some M ∈ R such that an < M for all n ∈ N.
there exists some M ∈ N such that an < M for all n ∈ N.
there exists some M ∈ N such that an ≥ M for all n ∈ N.
there exists some M ∈ R such that an ≥ M for some n ∈ N.
Define bounded
there exist M1, M2 ∈ R such that M1 ≤ an ≤ M2 for all n ∈ N.
there exist M1, M2 ∈ R such that M1 ≤ an ≤ M2 for some n ∈ N.
there exist M1, M2 ∈ N such that M1 ≤ an ≤ M2 for all n ∈ N.
there exist M1, M2 ∈ Q such that M1 ≤ an ≤ M2 for all n ∈ N.
there exist M1, M2 ∈ R such that M1 < an < M2 for all n ∈ N.
there exist M1, M2 ∈ R such that M1 ≤ an < M2 for all n ∈ N.
Give a sequence that is bounded but does not converge
an =
Lemma 1.9, Convergent sequences are bounded. Every sequence of numbers is a sequence
AOL: lim an = ℓ and lim bn = m Then,
lim(an + bn) = ?
ℓ + m,
ℓm,
ℓ - m
ℓ + m - e
AOL: lim an = ℓ Then,
lim λan = ?
λℓ
λ
ℓ
2λℓ
λ+ℓ
λ-ℓ
AOL: lim an = ℓ and lim bn = m Then, lim anbn = ?
ℓm
ℓ/m
ℓ + m
Sandwich Theorem/Squeeze Rule
. Let N ∈ N and ℓ ∈ R. Suppose (an), (bn) and (cn) are sequences satisfying an ≤ bn ≤ cn for all n ≥ N. If an → ℓ and cn → ℓ, then bn → ℓ.
. Let N ∈ N and ℓ ∈ R. Suppose (an), (bn) and (cn) are sequences satisfying an ≤ n ≤ cn for all n ≥ N. If an → ℓ and cn → ℓ, then bn → ℓ.
. Let N ∈ R and ℓ ∈ N. Suppose (an), (bn) and (cn) are sequences satisfying an ≤ bn ≤ cn for all n ≥ N. If an → ℓ and cn → ℓ, then bn → ℓ.
. Let N ∈ N and ℓ ∈ R. Suppose (an), (bn) and (cn) are sequences satisfying an ≤ bn ≤ cn for some n ≥ N. If an → ℓ and cn → ℓ, then bn → ℓ.
If |λ| < 1 then λ^n n → ? as n → ∞
0
1
n
∞
-∞
2
s>0 1/(n^s) → ? as n → ∞.
s
1/s
(n^s)/ n! → ? as n → ∞
n!
(λ^n)/n! → ? as n → ∞.
