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1st year Maths > RAC sequences and series > Flashcards

RAC sequences and series Flashcards

1
Q

sequence (aₙ)

A

an ordered of real numbers

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2
Q

series

A

the sum of all the terms in a sequence

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3
Q

trichotomy

A

exactly one of the following statements is true:

a<b></b>

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4
Q

a>b =>

A

a + c > b + c

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5
Q

if a>b and c>0 =>

A

ac > bc

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6
Q

if a>b and c<0 =>

A

ac < bc

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7
Q

a>b>0 =>

A

1/a < 1/b

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8
Q

if ab>0 <=>

A

a,b>0 or a,b<0

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9
Q

using a sign table

A
  1. split the expression
  2. decide which values cause any of the expressions to be 0
  3. write the sign for each expression and each value including the ‘ general’ inbetween values
  4. depending on whether or not its a strict inequality write the intervals which match
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10
Q

properties of the modulus function

A
  1. |x|>= 0 and |x| =0 <=> x=0
  2. since |x|^2 = x^2, so |x| =√(x^2)
    • |x|<=x<=|x|
  4. if Ɛ>0, |x| -Ɛ
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11
Q

|a-b|

A

the distance between a and b

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12
Q

triangle theorum

A

|a+b| <= |a| + |b| for all real numbers

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13
Q

Tending to infinity definition

A

A sequence (aₙ) tends to infinity if given any real number, A>0, there exists a point in the sequence, NЄℕ, such that an>A wherever n>N

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14
Q

Tending to minus infinity definition

A

A sequence (aₙ) tends to minus infinity if given any real number, A>0, there exists a point in the sequence, NЄℕ, such that anN

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15
Q

(aₙ) -> -∞ <=>

A

-(aₙ) -> ∞

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16
Q

Converging to a point definition

A

A sequence (aₙ) converges to a real number l if: given any small number, Ɛ>0, there exists a point in the sequence, NЄℕ, such that |(aₙ)-l|N

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17
Q

uniqueness of limits theorum

A

if a sequence converges then its limit, l, is unique

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18
Q

proving something tends to infinity

A
  1. Let A>0
  2. simplify the expression to observe that (expression for n)>A
  3. rearrange for n>(expression of A)
  4. provided n> expressions. choosing any natural number larger than expression we have an>A for all n>N
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19
Q

triangle theorem

A

|a+b| <= |a| + |b| for all real numbers

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20
Q

Suppose (an) converges to l>0. then there exists NЄℕ such that…

A

an> 0 whenever n>N

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21
Q

bounded above definition

A

there exists MЄℝ such that an<=M for all nЄℕ

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22
Q

bounded below definition

A

there exists MЄℝ such that an>=M for all nЄℕ

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23
Q

bounded definition

A

both bounded above and bounded below

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24
Q

if a sequence converges…

A

it is bounded. if a sequence is bounded it doesn’t mean it converges.

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25
Q

Algebra of limits theorem

A

suppose (an) and (bn) are sequences converging to l and M, then:

  1. an + bn -> l + M
  2. λan -> λl
  3. anbn -> lM
  4. an/bn -> l/M
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26
Q

Converging to a point definition

A

A sequence (aₙ) converges to a real number l if: given any small number, Ɛ>0, there exists a point in the sequence, NЄℕ, such that |(aₙ)-l|N

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27
Q

Tending to infinity definition

A

A sequence (aₙ) tends to infinity if given any real number, A>0, there exists a point in the sequence, NЄℕ, such that an>A wherever n>N

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28
Q

tending to minus infinity definition

A

A sequence (aₙ) tends to minus infinity if given any real number, A>0, there exists a point in the sequence, NЄℕ, such that anN

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29
Q

proving something converges to a limit

A
  1. let Ɛ>0
  2. sub in an and l into (aₙ)-l
  3. rearrange in terms of l
  4. Given N is any number at least as large as (expression in terms of Ɛ) we have (aₙ)-lN
30
Q

proving a complicated expression converges to a point

A
  1. (observe that) get in terms of 1/n for each value (remove other values if neccessary)
  2. using algebra of limits see what each term tends to
  3. work out the overall point of convergence
31
Q

what does 1/n converge to?

A

0

32
Q

proving algebra of limits theorems

A
  1. use convergence definition for each sequence/limit for all n>N1
  2. use triangle inequality
  3. take maximum of expression
33
Q

limiting limits of sequences proposition

A

an->l bn->m

if an<=bn then l

34
Q

sandwhich theorem

A

an<=bn<=cn

if an-> and cn -> l then bn -> l

35
Q

subsequence

A

selecting only certain terms from an original sequence

36
Q

convergence of a sub-sequence theorem

A

suppose an->l and ank is a susequence then ank->l

37
Q

showing a sequence doesn’t converge

A

if the sequence possesses subsequences converging to distinct limits then it does not converge

38
Q

Bolzano wierstrass Theorem

A

Every bounded sequence of real numbers includes a convergent subsequence

39
Q

Monotone increasing sequence

A

an+1>=an

40
Q

Monotone strictly increasing sequence

A

an+1>an

41
Q

Monotone decreasing sequence

A

an+1<=an

42
Q

Monotone strictly decreasing sequence

A

an+1

43
Q

Montone sequence

A

is either increasing, decreasing, strictly increasing or strictly decreasing

44
Q

Monotone convergence theorem

A
  1. if an is increasing and bounded above then it converges

2. if an is decreasing and bounded below then it converges

45
Q

eulers theorem

A

a sequence an given an=(1+1/n)^n converges

46
Q

A series is an expression of the form

A

Σaₙ = a₁+a₂+a₃+a₄+…

47
Q

method of differences

A
  1. split the expression into individual fractions
  2. subbing in values
  3. cancelling
  4. left with and expression in terms of N
48
Q

converging series

A

a series Σaₙ converges to a real number s if its sequence of partial sums Sₙ = Σaₙ converges to S

49
Q

diverges

A

if a series does not converge it diverges

50
Q

geometric series

A

Σrⁿ= 1 + r + r²+…+rⁿ

51
Q

sum of a geometric series

A

Sₙ = (1-r⁽ⁿ⁺¹⁾)/1-r if r!=0

52
Q

infinite sum of a geometri series

A

Sₙ = 1/(1-r) if |r|<1

53
Q

a geometric series converges iff

A

|r|<1

54
Q

Σ1/nᵃ converges

A

if a>1

55
Q

Algebra of limits for series theorem

A

suppose Σaₙ and Σbₙ converge and λ,μ are real then Σλaₙ + μbₙ converges and Σλaₙ + μbₙ = λΣaₙ + μΣbₙ

56
Q

Sequence Convergence Tests

A
  1. Null sequence test
  2. The comparison test
  3. The ratio test
  4. The root test
  5. The Integral test
  6. Alternating Series Test
  7. Absolute Convergence Test
57
Q

Null Sequence Test

A

if Σaₙ converges, then an->0

test for divergence only: if an->0 then Σaₙ diverges

58
Q

The Comparison Test

A

Suppose an>=0 bn>=an for n then:

  1. Σbₙ converges <=> Σaₙ converges
  2. Σaₙ diverges <=> Σbₙ diverges
59
Q

The Ratio Test

A 
Suppose Σaₙ is a series of non-negative terms and:
aₙ₊₁/aₙ -> r
then: 
1. if r<1 => Σaₙ converges
2. if r>1 Σaₙ  diverges
3. if r=0 inconclusive
60
Q

The root test

A

Suppose Σaₙ is a series of non-negative terms and aₙ¹/ⁿ->. Then:

  1. if r<1 => Σaₙ converges
  2. if r>1 => Σaₙ diverges
  3. if r=0 inconclusive
61
Q

Integral Test

A

Suppose f:[1,∞)-> is continuous, decreasing and positive in its domain. Then the series Σfₙ converges if and only if the sequence (∫f(x)dx) converges

62
Q

Alternating series test

A

Consider the series Σ(-1)ⁿ⁺¹aₙ where the sequence (aₙ) is decreasing and converging to zero. Then the series converges.

63
Q

Absolute Convergence Test

A

If a series converges absolutely absolutely, then it converges

64
Q

absolute convergence

A

Σaₙ converges absolutely if Σ|aₙ|

65
Q

conditional convergence

A

Σaₙ converges conditionally if it converges, but not absolutely

66
Q

rearrangement

A

we a sequence bₙ a rearrangement of a sequence an if there is a bijection σ:ℕ-> ℕ such that bₙ = aσ₍ₙ₎

67
Q

Dirichlets theorem (rearrangement of series)

A

Let Σaₙ be absolutely convergent. if bₙ is any rearrangement of aₙ then:

  1. Σbₙ is absolutely convergent
  2. Σbₙ = Σaₙ
68
Q

Conditional convergence theorem

A

suppose Σaₙ is conditionally convergent. Given γ there exists a rearrangement bₙ of aₙ such that Σbₙ=γ

69
Q

Multiplying series theorem

A

if Σaₙ and Σbₘ are Absolutely convergent then Σaₙbₘ is well-defined (order doesn’t matter) and absolutely convergent and: (Σaₙ) * (Σbₘ) = Σaₙbₘ

70
Q

form of a power series

A

Σaₙxⁿ

71
Q

convergence of power series theorem

A

Given a power series Σaₙxⁿ, either it converges absolutely for all x∈ℝ, or there exists ℝ∈[0,∞) such that

1st year Maths (24 decks)

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