Sequences and series- JB 1st term revision

a ≤ x ≤ b

{x ∈ R : a ≤ x ≤ b }

{x ∈ R : a < x < b }

a < x < b

(4,1)

Open Intervals

Closed Intervals

Closed Intervals

Open Intervals

|a + b| ≤ |a| + |b|

a + b ≤ |a| + |b|.

|a + b| < |a| + |b|

|a + b| ≤ |a| - |b|.

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

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.

∀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

x^2

x

|x|

-x

|x|

x

x^2

|x+1|

|x||y|

xy

|x+y|

|x|+|y|

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

1/e

e

2e

2/e

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

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.

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.

ℓ + m,

ℓm,

ℓ - m

ℓ + m - e

λℓ

λ

ℓ

2λℓ

λ+ℓ

λ-ℓ

ℓm

ℓ/m

ℓ + m

ℓ - m

. 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.

0

1

n

∞

-∞

2

λ

0

∞

n

s

1/s

0

n

n!

∞

λ

n!

0

∞

n