Sequences: Maths 1st Partial

What is a sequence

An ordered list of up to infinitely many numbers where f: ℕ₊ → ℝ

What are two features of a sequence as a set

• Composed of only isolated points

• Not open

a_m can also be written as

a_m = f(m)

Whats the formula for a sequence being bounded above

∃ k such that a_m < k ∀ m

What’s the formula for a sequence being bounded

∃ k such that |a_m| < k ∀ m

We say that {a_m} satisfies a property ___ if there is an integer m̄ such that a satisfies the property ___

We say that {a_m} satisfies a property eventually if there is an integer m̄ such that a satisfies the property ∀ m > m̄

the property is eventually satisfied if, past a certain location in the sequence, every point satisfies the property

How do we define monotone for a sequence?

a_m > a_m+1 ∀ m

Mathematical definition for how we say: {aₘ} is convergent to L ∈ ℝ if:

∀ ε > 0, ∃ m_ε such that m ≥ m_ε ⇒ |aₘ − L| < ε

there exists a number greater than zero such that for every number in the set a_m past a certain position m_ε, a_m is so close to L that a_m - L = this tiny number greater than 0, and therefore a_m is a sequence converging to L — often looks like the ‘tired frog’ sequence

Alternative neighbourhood based method:

We say that {aₘ} is convergent to L ∈ ℝ if:

∀ Bε(L), ∃ mε such that m ≥ mε ⇒ aₘ ∈ Bε(L)

for every small neighbourhood radius sigma around centre L, as you reach a certain point in the sequence, all numbers past that point in the sequence are an element of this tiny neighbourhood around the limit

We say that {aₘ} is positively divergent if:

∀ K, ∃ mₖ such that m ≥ mₖ ⇒ aₘ > K

for every large number k, there exists a certain location in the sequence where, past this point, every number in the sequence is greater than K

Neighbourhood-based definition of

We say that {aₘ} is positively divergent if:

∀ B(+∞), ∃ mₖ such that m ≥ mₖ ⇒ aₘ ∈ B(+∞)

For every neighbourhood of positive infinity (extending from K to +∞), there exists a position on the sequence past which every number is an element of this neighbourhood.

Give an example of a limit from above and how thats denoted

a_m = 1 / m

lim a_m = 0

Lim a_m = L⁺

What does ∀ B_ε⁺(L) refer to

[L, L + ε)

How do we mathematically define an a_m limited from above

∀ B_ε⁺(L), ∃ mε such that m ≥ mε ⇒ aₘ ∈ B_ε⁺(L)

( or equivalently:

aₘ ∈ [L, L + ε)

or:

L ≤ aₘ < L + ε )

Give an example of an a_m with a limit from below

a_m = 1 - 1/m

a frog that forever leaps closer and closer from 0 to 1 but never quite gets to 1

What is a_m when lim a_m is:

• L

• +∞

• -∞

• None

L — a_m convergent

+∞ — a_m positively divergent

-∞ — a_m negatively divergent

none — a_m oscillating or irregular

a_m is regular in the first three cases - when convergent or divergent

Give an example of osciallating bounded and oscillating unbounded

bounded: Sin(m)

unbounded: (-1)^m m²

What is implied by a_m being regular

It has a limit

What’s the logic behind lim a_m = +∞ ⇔ lim 1/a_m = 0⁺

a_m infinite ⇔ 1 / a_m infinitesimal

1 / 0⁺ = +∞

1 / +∞ = 0⁺

The reciprocal of a divergent sequence is equal to zero

What is called an infinite sequence and what is called an infinitesimal sequence

Infinite: lim a_m = +∞

Infinitesimal: lim b_m = 0

What does 0⁺ represent

0⁺ represents a limit where a positive variable approaches zero from above, and the limit evaluates to 0

a_m has a limit in R extended ⇒

the limit is unique

Convergent a_m ⇒

Convergent a_m ⇒bounded a_m

Monotone a_m ⇒

Monotone a_m ⇒ regular a_m

If a_m is monotone bounded ⇒

a_m monotone bounded ⇒ a_m convergent

lim a_m = L^- (approaching from below)

If a_m increasing unbounded ⇒

a_m increasing unbounded ⇒ a_m positively divergent

lim a_m = L

If a_m decreasing unbounded ⇒

a_m decreasing unbounded ⇒ a_m negatively divergent

What is lim m^a if a:

a > 0

a = 0

a < 0

lim m^a when a:

a > 0 +∞

a = 0 1

a < 0 0⁺

What is the limit of a geometric sequence lim q^m when q is:

q > 1

q = 1

-1 < q < 1

q = -1

q < -1

lim of q^m:

q > 1 +∞

q = 1 1

-1 < q < 1 0

q = -1 irregular bounded

q < -1 irregular unbounded

what is lim log_am when a:

a > 1

0 < a < 1

a > 1 +∞

0 < a < 1 -∞

What is lim (ln m)^a when alpha:

a > 0

a = 0

a < 0

a > 0 +∞

a = 0 1

a < 0 0⁺

lim a_m = A, lim b_m = B ^{_{(provided not indeterminate cases e.g. infinity minus infinity}}

What is lim (a_m + b_m)

lim (a_m + b_m) = A + B

lim (a_m • b_m)

lim (a_m • b_m) = A • B

lim (a_m / b_m)

lim (a_m / b_m) = A / B

How do we transform 0⁰ into regular notation using a^b = e^{loga ^b} = e^{b loga}

0⁰ transforms into e^0(-∞)

How do we transform +∞⁰ into regular notation using a^b = e^{loga ^b} = e^{b loga}

+∞⁰ = e^0(+∞)

How do we transform 1^∞ into regular notation using a^b = e^{loga ^b} = e^{b loga}

turns into e^{∞ ln1}

What is the comparison criterion

Let a < b_m < c_m be such that at least eventually

lim a_m = lim c_m = L

Then lim b_m = L

What is the ratio criterion

If lim | a_m+1 / a_m | = q with q < 1

Then lim a_m = 0

so to do this u take ur formula for a_m and substitute in (a_{m + 1}) then u just divide these two things by each other and mess around with the nummbers to find its limit and see if its less than 1, if it equals 1 then the test is inconclusive

What does lim (lower in the scale / higher in the scale) equal?

. lower in the scale

lim ————————— = 0

. higher in the scale

What is the order among infinities (when a_m, b_m → ∞) for lim (a_m / b_m)

How do ln m, m^a, and q^m relate to each other on a vague graph

What is the limit by comparing infinitesimals a_m,b_m → 0 for lim (a_m / b_m)

What do we call a_m when it is a “faster” infinitesimal than b_m

a_m is an infinitesimal of higher order than b_m

the exponential is a ___ limit than any power

the exponential is a faster limit than any power

When you have multiple infinitesimals to deal with, you first keep ___

the lower infinitesimals. the faster infinitesimals become negligible (e.g. 0.1 + 0.0000001 ~ 0.1)

lim ___ = e^a

lim (1 + a / m )^m = e^a

lim a_m / b_m = 1 means a_m ____ to b_m

a_m asymptotic to b_m

lim a_m / b_m = 0 means a_m ____ to b_m

lim a_m / b_m = 0 means a_m negligible with respect to b_m

when a_m ~ b_{m ,} a_m, b_m are ____

when a_m ~ b_{m ,} a_m, b_m are asymptotic

a_m and b_m are ‘quite’ equal

when a_m ~ b_m , lim(a_m / b_m) =

when a_m ~ b_m , lim(a_m / b_m) = 1

when a_m = o(b_m), a_m ____ to b_m

when a_m = o(b_m) little o b_m, a_m negligible with respect to b_m

when a_m = o(b_m) , lim(a_m / b_m) =

when a_m = o(b_m) , lim(a_m / b_m) = 0

What is a mathematical way of writing

a_m ~ b_m ⇔

a_m ~ b_m ⇔ a_m = b_m + o(b_m)

What is the formal mathematical definition for convergence of a sequence of vectors

lim xₘ = L

∀ ε > 0, ∃ m_ε such that m ≥ m_ε ⇒ d(xₘ, L) < ε

or equivalently:

∀ ε > 0, ∃ m_ε such that m ≥ m_ε ⇒ ‖xₘ − L‖ < ε

there exists a sigma where, past a certain point in the sequence a_mε , the distance between the vector points of the sequence and L is sigma, where sigma is close to zero