3 - Sequences and Convergence 2.0 Flashcards
Limit superior and limit inferior, subsequences, accumulation points, Cauchy sequences.
What is the Euler Number?
It is a limit value that forms a basis in logarithmic and exponential functions. The Euler Number sequence commonly takes the form:
e_n = (1 + 1/n)^n
It has other approximations
What is the limit value of the equation:
1 + 1/n
As n goes to infinity? Why?
One could argue that
1 + 1/n –> 1
As n –> infinity and then apply the limit laws to conclude that it converges to 1. Quite surprisingly this is not correct. The reason is that the limit laws only apply if the number of factors is constant. Here, however, the number of elements in e_n is n.
What is the arithmetic-geometric mean inequality?
The inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number).
Suppose that n ϵ N and that x1, ….. , xn >= 0. Then:
x1x2 … xn <= ([x1 + x2 + … + xn] / n) ^n
With strict inequality unless x1 = x2 = … = xn
What is a strict inequality?
A strict inequality exists in the same sense as a proper subset symbol. Meaning, a strict inequality is when a < b or b > a. As opposed to a <= b or b >= a. A strict inequality occurs when an element a is either greater or lesser than another element b. They are not equal.
What is the binomial formula?
It is a method used to expand large polynomial products, as follows:
^n C_k * p^n * (1-p)^n-k = (^n _k) [n! / (k! * (n-k)!)] * p^n * (1-p)^n-k
What is limit superior and inferior?
The limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence.
For a set, they are the infimum and supremum of the set’s limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them.
In other words;
limit superior = largest limit value
limit inferior = smallest limit value
What is an accumulation point?
Accumulation points of a sequence are the limits of convergent subsequences.
What are the accumulation points of the function:
x_n = (-1)^n
The sequence has two accumulation points, namely +1 and -1.
What are the limit superior and inferior in regards to accumulation points?
Limit inferior and superior are the smallest and largest accumulation points of a sequence
What is the Bolzano-Weierstrass theorem?
The Bolzano-Weierstrass theorem simply states that every bounded sequence has a convergent subsequence.
What is a Cauchy sequence?
A sequence of real numbers is called a Cauchy sequence if for every positive real number ε (ε > 0) there exists a positive integer n0 ϵ N such that:
|| xn - xm || < ε for all n,m > n0
In a similar way, one can define Cauchy sequences of rational or complex numbers.
Basically, if there’s an n0 such that every x >= x_n0 is within the ε-neighbourhood then it is a Cauchy sequence.
Are all convergent sequences Cauchy sequences?
A sequence is convergent if and only if it is a Cauchy sequence.
Are Cauchy sequences bounded or unbounded?
They are bounded
What is Bernoulli’s Inequality?
It is an inequality that approximates exponentiations of x + 1
Is the Euler number sequence monotone?
It is a strictly increasing sequence, therefore it is monotone.
