RAC sequences and series Flashcards
1
Q
sequence (aₙ)
A
an ordered of real numbers
2
Q
series
A
the sum of all the terms in a sequence
3
Q
trichotomy
A
exactly one of the following statements is true:
a<b></b>
4
Q
a>b =>
A
a + c > b + c
5
Q
if a>b and c>0 =>
A
ac > bc
6
Q
if a>b and c<0 =>
A
ac < bc
7
Q
a>b>0 =>
A
1/a < 1/b
8
Q
if ab>0 <=>
A
a,b>0 or a,b<0
9
Q
using a sign table
A
- split the expression
- decide which values cause any of the expressions to be 0
- write the sign for each expression and each value including the ‘ general’ inbetween values
- depending on whether or not its a strict inequality write the intervals which match
10
Q
properties of the modulus function
A
- |x|>= 0 and |x| =0 <=> x=0
- since |x|^2 = x^2, so |x| =√(x^2)
- |x|<=x<=|x|
- if Ɛ>0, |x| -Ɛ
11
Q
|a-b|
A
the distance between a and b
12
Q
triangle theorum
A
|a+b| <= |a| + |b| for all real numbers
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
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
15
Q
(aₙ) -> -∞ <=>
A
-(aₙ) -> ∞
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
17
Q
uniqueness of limits theorum
A
if a sequence converges then its limit, l, is unique
18
Q
proving something tends to infinity
A
- Let A>0
- simplify the expression to observe that (expression for n)>A
- rearrange for n>(expression of A)
- provided n> expressions. choosing any natural number larger than expression we have an>A for all n>N
19
Q
triangle theorem
A
|a+b| <= |a| + |b| for all real numbers
20
Q
Suppose (an) converges to l>0. then there exists NЄℕ such that…
A
an> 0 whenever n>N
21
Q
bounded above definition
A
there exists MЄℝ such that an<=M for all nЄℕ
22
Q
bounded below definition
A
there exists MЄℝ such that an>=M for all nЄℕ
23
Q
bounded definition
A
both bounded above and bounded below
24
Q
if a sequence converges…
A
it is bounded. if a sequence is bounded it doesn’t mean it converges.
25
Algebra of limits theorem
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
26
Converging to a point definition
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
27
Tending to infinity definition
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
28
tending to minus infinity definition
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
29
proving something converges to a limit
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
proving a complicated expression converges to a point
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
what does 1/n converge to?
0
32
proving algebra of limits theorems
1. use convergence definition for each sequence/limit for all n>N1
2. use triangle inequality
3. take maximum of expression
33
limiting limits of sequences proposition
an->l bn->m
| if an<=bn then l
34
sandwhich theorem
an<=bn<=cn
| if an-> and cn -> l then bn -> l
35
subsequence
selecting only certain terms from an original sequence
36
convergence of a sub-sequence theorem
suppose an->l and ank is a susequence then ank->l
37
showing a sequence doesn't converge
if the sequence possesses subsequences converging to distinct limits then it does not converge
38
Bolzano wierstrass Theorem
Every bounded sequence of real numbers includes a convergent subsequence
39
Monotone increasing sequence
an+1>=an
40
Monotone strictly increasing sequence
an+1>an
41
Monotone decreasing sequence
an+1<=an
42
Monotone strictly decreasing sequence
an+1
43
Montone sequence
is either increasing, decreasing, strictly increasing or strictly decreasing
44
Monotone convergence theorem
1. if an is increasing and bounded above then it converges
| 2. if an is decreasing and bounded below then it converges
45
eulers theorem
a sequence an given an=(1+1/n)^n converges
46
A series is an expression of the form
Σaₙ = a₁+a₂+a₃+a₄+...
47
method of differences
1. split the expression into individual fractions
2. subbing in values
3. cancelling
4. left with and expression in terms of N
48
converging series
a series Σaₙ converges to a real number s if its sequence of partial sums Sₙ = Σaₙ converges to S
49
diverges
if a series does not converge it diverges
50
geometric series
Σrⁿ= 1 + r + r²+...+rⁿ
51
sum of a geometric series
Sₙ = (1-r⁽ⁿ⁺¹⁾)/1-r if r!=0
52
infinite sum of a geometri series
Sₙ = 1/(1-r) if |r|<1
53
a geometric series converges iff
|r|<1
54
Σ1/nᵃ converges
if a>1
55
Algebra of limits for series theorem
suppose Σaₙ and Σbₙ converge and λ,μ are real then Σλaₙ + μbₙ converges and Σλaₙ + μbₙ = λΣaₙ + μΣbₙ
56
Sequence Convergence Tests
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
Null Sequence Test
if Σaₙ converges, then an->0
| test for divergence only: if an->0 then Σaₙ diverges
58
The Comparison Test
Suppose an>=0 bn>=an for n then:
1. Σbₙ converges <=> Σaₙ converges
2. Σaₙ diverges <=> Σbₙ diverges
59
The Ratio Test
```
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
The root test
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
Integral Test
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
Alternating series test
Consider the series Σ(-1)ⁿ⁺¹aₙ where the sequence (aₙ) is decreasing and converging to zero. Then the series converges.
63
Absolute Convergence Test
If a series converges absolutely absolutely, then it converges
64
absolute convergence
Σaₙ converges absolutely if Σ|aₙ|
65
conditional convergence
Σaₙ converges conditionally if it converges, but not absolutely
66
rearrangement
we a sequence bₙ a rearrangement of a sequence an if there is a bijection σ:ℕ-> ℕ such that bₙ = aσ₍ₙ₎
67
Dirichlets theorem (rearrangement of series)
Let Σaₙ be absolutely convergent. if bₙ is any rearrangement of aₙ then:
1. Σbₙ is absolutely convergent
2. Σbₙ = Σaₙ
68
Conditional convergence theorem
suppose Σaₙ is conditionally convergent. Given γ there exists a rearrangement bₙ of aₙ such that Σbₙ=γ
69
Multiplying series theorem
if Σaₙ and Σbₘ are Absolutely convergent then Σaₙbₘ is well-defined (order doesn't matter) and absolutely convergent and: (Σaₙ) * (Σbₘ) = Σaₙbₘ
70
form of a power series
Σaₙxⁿ
71
convergence of power series theorem
Given a power series Σaₙxⁿ, either it converges absolutely for all x∈ℝ, or there exists ℝ∈[0,∞) such that
