In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences), or the set of the first n natural numbers (for a sequence of finite length n). Sequences are one type of indexed families as an indexed family is defined as a function which domain is called the index set, and the elements of the index set are the indices for the elements of the function image.
For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...).
The position of an element in a sequence is its rank or index; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis, a sequence is often denoted by letters in the form of
a
n
{\displaystyle a_{n}}
,
b
n
{\displaystyle b_{n}}
and
c
n
{\displaystyle c_{n}}
, where the subscript n refers to the nth element of the sequence; for example, the nth element of the Fibonacci sequence
F
{\displaystyle F}
is generally denoted as
F
n
{\displaystyle F_{n}}
.
In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
The series that is given is $$\frac12+\frac13+\left(\frac12\right)^2+\left(\frac13\right)^2+\left(\frac12\right)^3+\left(\frac13\right)^3+\ldots.$$ Now, it's easy to see these are two separate geometric series, however, Spivak claims the ratio test fails because the ratio of successive terms...
Unfortunately, I have problems with the following task
For task 1, I proceeded as follows. Since the four bases have the same probability, this is ##P=\frac{1}{4}## I then simply used this probability in the formula for the Shannon entropy...
How do I approach the following problem while only knowing the PSD of a Gaussian random sequence (i.e. I don't know the exact distribution of $V_k$)? Or am I missing something obvious?
Problem statement:
Thoughts:
I know with the PSD given, the autocorrelation function are delta functions due...
I've been given the proof, but don't understand; to calculate the limit of ##f## when ##x## tends to zero it's enough to see that if ##\{x_n\}_{n=1}^\infty## is a sequence that tends to ##0##, then...
Consider the following:
We start with a positive integer: x
If x is even, do
x/2
If is odd, do
Floor function( x * y)
with y being some decimal number between 1 and 2
And repeat until a loop is reached. If 1 is reached, the next number will be 1 as well. So we reach a loop too.
An example:
x =...
Hello all, I have a random sequences question and I am mostly struggling with the last part (e) with deriving the marginal pdf. Any help would be greatly appreciated.
My attempt for the other parts a - d is also below, and it would nice if I can get the answers checked to ensure I'm...
Prove the existence of a strictly increasing sequence ##m_1 < m_2 < m_3 < \cdots## of integers satisfying the property that for all positive integers ##\ell##, the sequence ##\sin(\ell m_1), \sin(\ell m_2), \sin (\ell m_3),\ldots## converges.
I have tried inserting 0.955 in the above formula for the sum of a geometric series and setting it equal to 3,000,000 (S_n) with n =3. This did not work out well
My second attempt was, considering that the payment is paid every year in the future, to use the convergence formula. There k = 0.955...
Homework Statement:: Tell me if a sequence or series diverges or converges
Relevant Equations:: Geometric series, Telescoping series, Sequences.
If I have a sequence equation can I tell if it converges or diverges by taking its limit or plugging in numbers to see what it goes too?
Also if I...
I would like some clarity on the highlighted part. My question is, consider the the attached example ##(c)##, This sequence converges ( by using L'Hopital's rule)...now my question is, the sequence is indicated on text as not being monotonic...very clear. Does it imply that if a sequence is not...
Let ##\omega_1## be the first uncountable ordinal such that ##x## is an element of ##\omega_1## if and only if it is either a finite ordinal or there exists a bijection from ##x## onto ##\omega##.
I want to define a matrix such that the matrix contains each element of ##\omega_1## only once.
To...
I don't understand what the question is asking.
the nth term of the first sequence i can calculate to be -2n+4, while 2n-24 is the nth term for the second sequence. now what? The question isn't clear.
I've started by writing down the definitions, so we have
$$x_n-y_n\rightarrow 0\, \Rightarrow \, \forall w>0, \exists \, n_w\in\mathbb{N}:n>n_{w}\,\Rightarrow\,|x_n-y_n|<w $$
$$(x_n)\, \text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|x_m-x_n|<w...
Upper bound definition for sets: $ M \in \mathbb{R} $ is an upper bound of set $ A $ if $ \forall \alpha\in A. \alpha \leq M$
Upper bound definition for sequences: $ M \in \mathbb{R} $ is an upper bound of sequence $ (a_n)$ if $ \forall n \in \mathbb{N}. a_n \leq M$
Suppose we look at the...
I refer to this page: https://taoanalysis.wordpress.com/2020/03/26/exercise-5-3-2/
I am having trouble understanding the purpose / motivation behind using the min as in ##\delta := \min\left(\frac{\varepsilon}{3M_1}, 1\right)## and ##\varepsilon' := \min\left(\frac{\varepsilon}{3M_2}...
Let us just lay down some definitions. Both sequences are equivalent iff for each ##\epsilon>0## , there exists an N>0 such that for all n>N, ##|a_n-b_n|<\epsilon##.
A sequence is a Cauchy sequence iff ##\forall\epsilon>0:(\exists N>0: (\forall j,k>N:|a_j-a_k|>\epsilon))##.
We proceeded by...
I was just reading about DNA sequencing. In my view, DNA can be modeled into an ordered sequence of nucleobases, as if the two strands were joined into a single strand (just like in RNA). The first half of the sequence models the first strand. The four nucleobases are numbered from 0 to 3...
Summary:: Sequences, Progressions
Hello. I have been Given the following exercise, Let (a1, a2, ... an, ..., a2n) be an arithmetic progression such that the sum of the last n terms is equal to three times the sum of the first n terms. Determine the sum of the first 10 terms as a function of...
Hello there.Is there any function or sequence that has no limits at any point? I am not necessarily talking about functions on euclidean spaces, they could be on topological spaces in general.Also, we have homeomorphism that is about I think mostly continuity, diffeomorphism about...
A sequence is made up of two sequences
an=(n^2)/(n+2) - (n^2)/(n+3)
The problem asks for the solver to work out if it's converging or diverging, and find a limit if possible.
My first thought was to write both over a common denominator and then divide through by the dominant term; this...
The book is Calculus: Basic Concepts for High School
on the first page you are given the following sequence:
1, -1, 1/3, -1/3, 1/5, -1/5, 1/7, -1/7, ...
several pages later the rule is given:
in the second rule, for the first term in the sequence, the coefficient of one of the terms is 1/0...
Of the various notions of convergence for sequences of functions (e.g. pointwise, uniform, convergence in distribution, etc.) which of them can describe convergence of a sequence of functions that have different domains? For example, let ##F_n(x)## be defined by ##F_n(x) = 1 + h## where ##h...
Hi,
A person has 40 litres of milk. As soon as he sells half a litre, he mixes the remainder with half a litre of water. How often can he repeat the process, before the amount of milk in the mixture is 50% of the whole?
Detailed explanation is appreciated.:)
Solution:
I am working on...
Hey! 😊
Between the following two topics:
Elementary Geometry
Fibonacci and its sequences
which would you suggest for a presentation? Could you give me also some ideas what could we the structure of each topic? :unsure:
The number sequence is as follows:
3x+5y; 5x; 7x-5y; 9x-10y...
I need to formulate a general term - Tn=T1+d(n-1)
In the above sequence I have no idea what.
I also think this sequence is non linear.
Please help with a solution
Thanks
I need help only in section 3
I have some kind of solution but I'm not sure because it seems too short and too simple.
We showed in section 1 that an> 0 per n.
Given that an + 1 <0 and an + 1 = an / a1 therefore a1 <0 is warranted
Hey! :o
Check the below sequences for convergence and determine the limit if they exist. Justify the answer.
$\displaystyle{f_n:=\left (1-\frac{1}{2n}\right )^{3n+1}}$
$\displaystyle{g_n:=(-1)^n+\frac{\sin n}{n}}$
I have done the following:
$\displaystyle{f_n:=\left...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...
I need yet further help in fully understanding the proof of Proposition 8.7 ...Proposition...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ...
I need some further help in fully understanding the proof of Proposition 8.7 ...Proposition...
If n is ∞, then ln (n) = ln (∞) = ∞
Then, 1/∞ = 0
Any number raised to "0" = 1, so the answer should be 1. However the book says the answer is e2. Could you provide me some help?
I know that human genome sequencing was done by 2001. Inintially there were Maxam-Gilberth technique, then Sanger's technique and then finally NGS techniques have made sequening faster and efficient. My question is, though we are able to sequence the whole genome, how have scientists arrived at...
I am reading N. L. Carothers' book: "Real Analysis". ... ...
I am focused on Chapter 3: Metrics and Norms ... ...
I need help with a remark by Carothers concerning convergent sequences in \mathbb{R}^n ...Now ... on page 47 Carothers writes the following:
In the above text from Carothers we...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with an aspect of Example 2.3.52 ...
The start of Example 2.3.52 reads as follows ... ...
In the above Example from Sohrab we...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...
I am focused on Chapter 3: Convergent Sequences ... ...
I need some help to fully understand the proof of Corollary 3.2.7 ...Garling's statement and proof of...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...
I am focused on Chapter 3: Convergent Sequences
I need some help to fully understand the proof of Corollary 3.2.7 ...Garling's statement and proof of...
I consider two sequences of numbers $A=\{a_1,...,a_n\}$ and $B=\{k-a_1,...,k-a_n\}$, where $a_1 \le a_2 \le ... \le a_n \le k$.
I am looking for such conditions under which: $gcd(a_1,...,a_n) = gcd(k-a_1,...,k-a_n)=1$.
In more general form: $gcd(a_1,...,a_n) = gcd(k-a_1,...,k-a_n) \ge 1$.
I...
Please can anyone help with the below problem? It’s an interesting problem but please bear with me as i don't have much math background.
A factory’s product is sampled once per month every month by its quality inspection team. The factory is allowed up to 2 product failures per ROLLING 12...
I am currently studying Fisher's formalism as part of parameter estimation.
From this documentation :
They that Fisher matrix is the inverse matrix of the covariance matrix. Initially, one builds a matrix "full" that takes into account all the parameters.
1) Projection : We can then do...
Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...
Homework Statement
Let ##a## and ##b## be positive numbers with ##a \gt b##. Let ##a_1## be their arithmetic mean and ##b_1## their geometric mean:
##a_1 = \frac {a + b} 2## and ##b_1 = \sqrt{ab}##
Repeat this process so that, in general
##a_{n + 1} = \frac {a_n + b_n} 2## and ##b_{n + 1} =...
I understand that when a sequence is described recursively, for example: ##a_1=2, a_{n+1} = \sqrt{3a_n}## then we mean that the first term is 2, the second term is ##\sqrt{3*2} = \sqrt{6}##, the third term is ##\sqrt{3*\sqrt{6}}##, and so on.
What I do not understand is how to interpret the...
Homework Statement
Identify the boundary ##\partial c_{00}## in ##\ell^p##, for each ##p\in[1,\infty]##
Homework Equations
The interior of ##S## is ##\operatorname{int}(S) = \{a\in S \mid \exists \delta > 0 \text{ such that } B_\delta (a) \subseteq S\}##.
##\partial S = \bar{S}\setminus...
In the thread https://www.physicsforums.com/threads/recursive-square-root-inside-square-root-problem.954655/ a sequence interpreted from the notation:
##\large{\sqrt{2+\pi \sqrt{3+\pi\sqrt{4+\pi\sqrt{5+\dotsb}}}}}##
was discussed.
What sequence would you (fellow forum members) associate with...
Homework Statement
For each ##n\in\mathbb{N}##, let the finite sequence ##\{b_{n,m}\}_{m=1}^n\subset(0,\infty)## be given. Assume, for each ##n\in\mathbb{N}##, that ##b_{n,1}+b_{n,2}+\cdots+b_{n,n}=1##.
Show that ##\lim_{n\to\infty}( b_{n,1}\cdot a_1+b_{n,2}\cdot a_2+\cdots+b_{n,n}\cdot a_n) =...
Homework Statement
Prove or produce a counterexample: If ##\{a_n\}_{n=1}^\infty\subset \mathbb{R}## is convergent, then
##
\min (\{a_n:n\in\mathbb{N}\} )## and
##\max (\{a_n:n\in\mathbb{N}\} )
##
both exist.
Homework EquationsThe Attempt at a Solution
I will produce a counterexample...
I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it.
Problem:
Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence $(a_n)$ of integers...
Homework Statement
Determine the triangles where the sides are consecutive elements of a geometric sequence and the angles are consecutive elements of an arithmetic sequence.
Homework Equations
The Attempt at a Solution
I don't really know how to approach this problem, what the solution would...
I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...
Homework Statement
Let ##K\neq\emptyset## be a compact set in ##\Bbb{R}## and let ##c\in\Bbb{R}##. Then ##\exists a\in K## such that ##\vert c-a\vert=\inf\{\vert c-x\vert : x\in K\}##.
2. Relevant results
Any set ##K## is compact in ##\Bbb{R}## if and only if every sequence in ##K## has a...