2 - Sequences and Convergence Flashcards
Limit laws, squeeze law, monotone convergence theorem, convergence tests. Euler number, n-th root.
What is a sequence?
Informally, a sequence in a set X is an infinite list of elements in X:
x1, x2, x3, ….
Formally, a sequence in a set X is a function x: N –> X
Is a sequence a function or a list?
Both. It is informally noted as a list, but formally as a function.
What are some examples of sequences?
- A constant sequence: a, a, a, a, …
- The sequence of even N numbers: 2, 4, 6, 8, 10, … (x_n = 2n)
- The sequence of prime numbers: 2, 3, 5, 7, 13, 17, 19, …
- The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …
- a_n = 1/n: 1, 1/2, 1/3, 1/4, …
- a_n = (-1)^n = -1, 1, -1, 1, -1, …
- 0, 1, -1, 2, -2, 3, -3, 4, -4, …
- a_n = i^n: i, -1, -i, 1, i, -1, -i, 1
Can sequences be bounded?
Yes.
A sequence (x_n) in R is bounded from above if there is M ϵ R such that x_n <= M for all n ϵ N.
It is bounded from below if there is m ϵ R such that x_n >= m for all n ϵ N.
It is called bounded if it is bounded from above and below.
Can complex sequences be bounded?
Yes. A sequence (x_n) in C is bounded if there is M > 0 such that |x_n| <= M for all n ϵ N.
What are some examples of partially or fully bounded sequences
- A constant sequence: a, a, a, a, …. Bounded
- The sequence of even N numbers: 2, 4, 6, 8, 10, … (x_n = 2n). Bounded from below by 2
- The sequence of prime numbers: 2, 3, 5, 7, 13, 17, 19, …. Bounded from below by 2
- The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …. Bounded from below by 1
- Harmonic series: a_n = 1/n: 1, 1/2, 1/3, 1/4, … Bounded
- Alternating series: a_n = (-1)^n = -1, 1, -1, 1, -1, … Bounded
What is convergence?
We say that (x_n) converges to x as n goes to infinity if for every epsilon > 0 there exists n0 ϵ N such that |x_n - x_n0| < ϵ for all n > n_0.
If (x_n) does not converge to any value, we say that it is divergent.
Can sequences converge?
Yep
Do sequences have limits?
Yes, convergent sequences have proper limits denoted as:
x = lim (n –> ∞) x_n
Does a sequnce always either converge or diverge?
Yes. There are no other options. It is either convergent or divergent.
What is an epsilon neighbourhood?
The set of all points in a metric space whose distance from a given point is less than some number; this number is designated ε.
Do constant sequences have convergence or divergence?
Yes. All constant sequences converge to the real-numbered term in the sequence.
Are convergent sequences bounded?
Each convergent sequence is bounded. But not every bounded sequence is convergent.
Are unbounded sequences divergent or convergent?
They are divergent.
What are the limit laws?
Let (x_n), (y_n) be convergent sequences, lim x_n = x, lim y_n = y.
Then:
- (x_n + y_n) is convergent, lim (x_n + y _n) = x + y
- (x_ny_n) is convergent, lim x_ny_n =x*y
- If x ≠ 0 then lim 1/x_n = 1/x
What is the Squeeze Law?
If two functions squeeze together at a particular point, then any function trapped between them will get squeezed to that same point. The squeeze theorem deals with limit values, rather than function values. The squeeze theorem is sometimes called the sandwich theorem or the pinch theorem.
Suppose that (a_n), (b_n), and (x_n) are sequences in R such that there exists m ϵ N, where a_n <= x_n <= b_n for all n >= m. If a_n –> x and b_n –> x, then (x_n) also converges to the limit value x.
What is monotonicity?
Let (a_n) be a sequence in R.
- a_n is increasing if a_n <= a_n+1 for all n ϵ N
- a_n is strictly increasing if a_n < a_n+1 for all n ϵ N
- a_n is decreasing if a_n >= a_n+1 for all n ϵ N
- a_n is strictly decreasing if a_n > a_n+1 for all n ϵ N
In each of these cases, we say that the sequence (a_n) is (strictly) monotone.
Do monotone and bounded sequences have convergence or divergence?
Every monotone and bounded sequence is convergent.
What are the limits of unbounded sequences?
If for every M ϵ R there is n_0 ϵ N such that a_n > M for all n > n_0 = n_0 (M). I.e. if it is an increasing sequence. Then the limit of the sequence is positive infinity. In other words, if there’s always a bigger number it goes to positive infinity.
If for every M ϵ R there is n_0 ϵ N such that a_n < M for all n > n_0 = n_0 (M). I.e. if it is a decreasing sequence. Then the limit of the sequence is negative infinity. In other words, if there’s always a smaller number it goes to negative infinity.
What is the ratio test?
The ratio test is a most useful test for series convergence. It caries over intuition from geometric series to more general series. It helps to determine whether or not a sequence/series is convergent or divergent. It works like this; suppose we have a series (a_n). Define:
L = lim | a_n+1 / a_n |
Then,
- If L<1 the series is absolutely convergent (and hence convergent)
- If L>1 the series is divergent.
- If L=1 the series may be divergent, conditionally convergent, or absolutely convergent.
Given:
C = { x | x ϵ Q, (x + (2)^0.5)^2 <= 2 }
What are the upper and lower bounds of the set? What is the maximum, minimum, supremum and infimum?
Upper bound: [0, +infinity)
Lower bound: (-infinity, 0]
Maximum: 0
Supremum: 0
Minimum: Does not exist
Infimum: -2*(2)^0.5
Denote A and B as sets, both bounded from below. Given A ⊂ B, prove that inf A >= inf B.
Let: α = inf A, β = inf B
β <= b for all b ϵ B
β <= a for all a ϵ A, since A ⊂ B
Therefore, β is an lower bound of A
Therefore;
inf B <= inf A
What is the preservation of inequalities?
Let (a_n) and (b_n) be sequences in R with a_n –> a and b_n –> b. If there exists m ϵ N such that a_n <= b_n for all n >= m, then a <= b.
Does a sequence have to converge always to be called convergent?
In order for a sequence to converge, it only needs to be bounded and monotne eventually.
What are improper limits?
Improper limits are when the limit value of a sequence is either positive or negative infinity. The limit laws cannot be applied if such improper limits are involved.
What are proper limits?
Limit values that are not positive or negative infinity
