# What is Sequences: Definition and 586 Discussions

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.

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1. ### I How does the ratio test fail and the root test succeed here?

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...
2. ### Calculating Shannon Entropy of DNA Sequences

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...
3. ### Probability involving Gaussian random sequences

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...
4. ### Why does this function make it easy to prove continuity with sequences?

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...
5. ### B Golden Ratio in Collatz-like sequences

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 =...
6. ### Sinusoidal sequences with random phases

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...
7. ### POTW Infinite Sequences of Sines

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.
8. ### Trust Fund problem using series and sequences

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...
9. ### B Understanding about Sequences and Series

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...
10. ### Bounded and monotonic sequences - Convergence

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...
11. ### I Limits of Two Ordinal Sequences

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...
12. ### Calculating nth Term of Sequences: What Now?

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.
13. ### Do Cauchy Sequences Imply Convergent Differences?

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...
14. ### MHB Upper Bound of Sets and Sequences: Analyzing Logic

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...
15. ### Understanding the Use of Min in Cauchy Sequences

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}...
16. ### Proof that two equivalent sequences are both Cauchy sequences

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...
17. ### DNA sequencing and restoring malformed sequences

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...
18. ### How to Solve Arithmetic Sequence Problems

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...
19. T

### I Limits of functions and sequences

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...
20. ### Converging Vs diverging sequences

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...
21. ### B Weird stuff on infinite numerical sequences in a Soviet book

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...
22. ### I Convergence of sequences of functions with differing domains?

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...
23. ### MHB Solve Math Problem: Mixing Milk & Water to Get 50% Milk

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...
24. ### MHB Topic of presentation: Elementary Geometry vs Fibonacci & its sequences

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:
25. ### MHB Number patterns and sequences - Tn Term

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
26. ### Proving inequalities for these Sequences

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

48. ### Triangle determined by arithmetic and geometric sequences

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...
49. ### I Question regarding a sequence proof from a book

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...
50. ### Distance of a point from a compact set in ##\Bbb{R}##

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...