2 - Sequences and Convergence

Question 1

What is a sequence?

Answer 1

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

Question 2

Is a sequence a function or a list?

Answer 2

Both. It is informally noted as a list, but formally as a function.

Question 3

What are some examples of sequences?

Answer 3

1. A constant sequence: a, a, a, a, …
2. The sequence of even N numbers: 2, 4, 6, 8, 10, … (x_n = 2n)
3. The sequence of prime numbers: 2, 3, 5, 7, 13, 17, 19, …
4. The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …
5. a_n = 1/n: 1, 1/2, 1/3, 1/4, …
6. a_n = (-1)^n = -1, 1, -1, 1, -1, …
7. 0, 1, -1, 2, -2, 3, -3, 4, -4, …
8. a_n = i^n: i, -1, -i, 1, i, -1, -i, 1

Question 4

Can sequences be bounded?

Answer 4

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.

Question 5

Can complex sequences be bounded?

Answer 5

Yes. A sequence (x_n) in C is bounded if there is M > 0 such that |x_n| <= M for all n ϵ N.

Question 6

What are some examples of partially or fully bounded sequences

Answer 6

1. A constant sequence: a, a, a, a, …. Bounded
2. The sequence of even N numbers: 2, 4, 6, 8, 10, … (x_n = 2n). Bounded from below by 2
3. The sequence of prime numbers: 2, 3, 5, 7, 13, 17, 19, …. Bounded from below by 2
4. The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …. Bounded from below by 1
5. Harmonic series: a_n = 1/n: 1, 1/2, 1/3, 1/4, … Bounded
6. Alternating series: a_n = (-1)^n = -1, 1, -1, 1, -1, … Bounded

Question 7

What is convergence?

Answer 7

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.

Question 8

Can sequences converge?

Answer 8

Yep

Question 9

Do sequences have limits?

Answer 9

Yes, convergent sequences have proper limits denoted as:

x = lim (n –> ∞) x_n

Question 10

Does a sequnce always either converge or diverge?

Answer 10

Yes. There are no other options. It is either convergent or divergent.

Question 11

What is an epsilon neighbourhood?

Answer 11

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.

Question 12

Do constant sequences have convergence or divergence?

Answer 12

Yes. All constant sequences converge to the real-numbered term in the sequence.

Question 13

Are convergent sequences bounded?

Answer 13

Each convergent sequence is bounded. But not every bounded sequence is convergent.

Question 14

Are unbounded sequences divergent or convergent?

Answer 14

They are divergent.

Question 15

What are the limit laws?

Answer 15

Let (x_n), (y_n) be convergent sequences, lim x_n = x, lim y_n = y.

Then:

1. (x_n + y_n) is convergent, lim (x_n + y _n) = x + y
2. (x_ny_n) is convergent, lim x_ny_n =x*y
3. If x ≠ 0 then lim 1/x_n = 1/x

Question 16

What is the Squeeze Law?

Answer 16

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.

Question 17

What is monotonicity?

Answer 17

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.

Question 18

Do monotone and bounded sequences have convergence or divergence?

Answer 18

Every monotone and bounded sequence is convergent.

Question 19

What are the limits of unbounded sequences?

Answer 19

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.

Question 20

What is the ratio test?

Answer 20

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,

1. If L<1 the series is absolutely convergent (and hence convergent)
2. If L>1 the series is divergent.
3. If L=1 the series may be divergent, conditionally convergent, or absolutely convergent.

Question 21

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?

Answer 21

Upper bound: [0, +infinity)
Lower bound: (-infinity, 0]
Maximum: 0
Supremum: 0
Minimum: Does not exist
Infimum: -2*(2)^0.5

Question 22

Denote A and B as sets, both bounded from below. Given A ⊂ B, prove that inf A >= inf B.

Answer 22

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

Question 23

What is the preservation of inequalities?

Answer 23

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.

Question 24

Does a sequence have to converge always to be called convergent?

Answer 24

In order for a sequence to converge, it only needs to be bounded and monotne eventually.

Question 25

What are improper limits?

Answer 25

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.

Question 26

What are proper limits?

Answer 26

Limit values that are not positive or negative infinity