# What is Sequence and series: Definition and 21 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. ### Doubt regarding the series ##\sum [\sqrt{n+1} - \sqrt{n}\,]##

We're given the series ##\sum_{n=1}^{\infty} [ \sqrt{n+1} - \sqrt{n} ]##. ##s_n = \sqrt{n+1} - 1## ##s_n## is, of course, an increasing sequence, and unbounded, given any ##M \gt 0##, we have ##N = M^2 +2M## such that ##n \gt N \implies s_n \gt M##. Thus, the series must be divergent. But...
2. ### 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...
3. ### Finding the convergence of a parametric series

It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with...
4. ### I ##(a_n) ## has +10,-10 as partial limits. Then 0 is also a partial limit

Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##. Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...
5. ### I Do I need induction to prove that this sequence is monotonic?

I think the initial assumptions would allow me to prove this without induction. Suppose ##(x_n)## is a real sequence that is bounded above. Define $$y_n = \sup\{x_j | j \geq n\}.$$ Let ##n \in \mathbb{N}##. Then for all ##j \in \mathbb{N}## such that ##j \geq n + 1 > n## $$x_{j} \leq y_n.$$...
6. ### Find the sum of the series Σ((3n+2)/n).... (confirmation)

Hi, this is my try:Thanks.
7. ### Sum of a series that tends to infinity

I tried by ##S=1+(1/1!)(1/4)+(1.3/2!)(1/4)^2+...## ##S/4=1/4+(1/1!)(1/4)^2+(1.3/2!)(1/4)^3..## And then subtracting the two equations but i arrived at nothing What shall i do further?
8. ### MHB Sum of pqth Term in Arithmetic Progression

The pth term of an airthmetic progression is 1/q and qth term is 1/p. What is the sum of pqth term?
9. ### Sequence and Series Terminology

First, I would like to clear up notation and the definition for sequences. What exactly is a sequence? I read somewhere that it is defined as a function ##f: \mathbb{N} \to \mathbb{R}##. But if this is the case, why do we only define functions based on the range of the function, e.g., ##\left \{...
10. ### What is the sum of cubes in a series up to 'n' terms?

Homework Statement find the sum of the series to 'n' terms (1^3 / 1 ) + (1^3 + 2^3 / 1+2 ) + (1^3 + 2^3 + 3^3 / 1+2+3 ) +... Homework Equations sigma n^2 = n(n+1)(2n+1) / 6 sigma n = n(n+1) / 2 The Attempt at a Solution numerator = 1^3 + 2^3 + ... n^3 denominator = 1+2+3+...n...
11. ### Sequence and series - Arithmetic mean question have been breaking my head

Homework Statement Two consecutive numbers from 1,2,3...n are removed A.M of remaining numbers is 105/4. Find n and those numbers removed .Homework Equations Answer n = 50 those numbers are 7 and 8 The Attempt at a Solution I solved this question like a few weeks ago but now it escaped my...
12. ### Definition of sequence and series

Hi! While studying sequence and series, I've gotten some misunderstandings in the definitions of sequence and series. What I know about the definitions of sequence and series is as follows below ; a sequence of field of real numbers is defined as a function mapping of the set of all positive...
13. ### Sequence and Series question (maybe)?

Homework Statement Find all positive integers n such that both n + 2008 divides n^2 + 2008 and n + 2009 divides n^2 + 2009 Homework Equations - The Attempt at a Solution I have no idea where to start... I'm not even sure it's a sequence and series question. If it is then I have...
14. ### Sequence and Series Reconstruction

After studying Generating Functions some, I wondered this question: 1). Take an arbitrary sequence of numbers for example: {p1,p2,p3,p4...pN} and get/find a Generating Function for that sequence. 2). Now 'remove' an arbitrary sequence members from the above set :{p1,p2,p3,p4...p100}...
15. ### Representing Sequence 1*3*5*7*9... with Factorials for Calculator Use

is there a way to represent the sequence 1*3*5*7*9...(2n-1) for n>=1 with a factorial if not is thre a way to input that sequence ino a calculator with somthing
16. ### How Do You Solve Challenging Limits and Analyze Sequence Monotonicity?

Got a couple questions please help! 1. lim n>inf (x^4)/(e^(-2x)) How do you do this one? i know you use L hospital rule but you can never get rid of the denominator. or is there a trick i am missing? 2. Determine whether the sequences are increasing, decreasing, or not monotonic...
17. ### Solving Sequence and Series Limits: xn = 1/ln(n+1)

Homework Statement Let xn = 1/ln(n+1) for n in N. a) Use the definition of limit to show that lim(xn) = 0. b) Find a specific value of K(ε) as required in the definition of limit for each of i)ε=1/2, and ii)ε=1/10. The Attempt at a Solution a) If ε > 0 is given, 1/ln(n+1) < ε <=>...
18. ### Sequence and Series, finding relationship

Homework Statement 0<y0<x0 x1=(x0+y0)/2 y1=\sqrt{x0y0} in general Xn+1=(xn+yn)/2 Yn+1=\sqrt{XnYn} Homework Equations none The Attempt at a Solution I have no idea I tried to solve for Xn and substituting that into another equation... however, I don't know how to simplify...
19. ### Expert Sequence and Series Assistance | Dave's Math Help

There was this awesome help site for this topic. The name of it was something like "Dave's math help". I thought I saved the link but guess I didn't. Anyone know where I could possibly find this site or know of others?
20. ### Sequence and series word problem

Tod injured his wrist while he was skateboarding. To relive the pain his doctor prescribed him some medication. He takes 500mg of medicine every 6h. Only 25 % of the medication remains in his body by the time he is ready to take another pill. Write and equation to model this. so 25% is...
21. ### Proving Cube Numbers Not in Sequence a_n=3n+2

Right here is my sequence 2, 5, 8, 11, 14, ... I have been asked to prove that the cube of any number in the sequence is in the sequence. my answer: General term: a_n=3n+2 We need to cube a_n and see if it matches a number in the series i.e. (a_n)^3 = 3q+2 for some integer q...