What is Sequences and series: Definition and 58 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. Vividly

    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...
  2. BerriesAndCream

    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.
  3. Purpleshinyrock

    Arithmetic sequences Problem

    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...
  4. WMDhamnekar

    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...
  5. JD_PM

    I Understanding why ##(y_n)_n## is a bounded sequence

    Suppose ##(y_n)_n## is a sequence in ##\mathbb{C}## with the following property: for each sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, also the series ##\sum_n \left(x_ny_n\right)## converges absolutely. Can you then conclude that ##(y_n)_n##...
  6. Gionata

    I Recursive square root inside square root problem

    I have been debating this issue for days: I can't find a recursive function of this equation: ##\large{\sqrt{2+\pi \sqrt{3+\pi\sqrt{4+\pi\sqrt{5+\dotsb}}}}}## Starting value 2 always added with pi has been trying to find a solution this for days now, is what I have achieved so far: This...
  7. L

    A Convergence of a subsequence of a sum of iid r.v.s

    ##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##. Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...
  8. H

    ##\lim_{n \to \infty} ## for the sequence at ##a_n##

    Homework Statement ##a_1## = 3 ##a_{n+1}## = ##\frac{2}{3} a_n + \frac{1}{4} ## Homework Equations [/B]The Attempt at a Solution Sequence i got : ## a_1, a_2, a_3, a_4, a_5 ## ## 3, \frac{9}{4}, \frac{7}{4}, \frac{17}{12}, \frac{43}{36} ## I tried to find the formula of ##a_n## ##a_n =...
  9. ertagon2

    MHB Solve Maths Sequences & Series - Get Passing Grade Now

    Can someone help me with these. These are the last 5 questions that I have to do and if I get them right I pass maths.
  10. N

    Expand x^(k+1)/(k+1) - (x-1)^(k+1)/(k+1)

    Homework Statement Expand x(k+1)/(k+1) - (x-1)(k+1)/(k+1) Homework Equations (a+b)m = am + mam - 1b + (mℂ2)am - 2b2 + ... + bm[/B] The Attempt at a Solution Here is my solution, I would like to know if it's correct or not I have the solution in an attached image
  11. E

    A Measuring the degree of convergence of a stochastic process

    Consider a sample consisting of {y1,y2,...,yk} realisations of a random variable Y, and let S(k) denote the variance of the sample as a function of its size; that is S(k)=1/k( ∑ki=1(yi−y¯)2) for y¯=1/k( ∑ki=1 yi) I do not know the distribution of Y, but I do know that S(k) tends to zero as k...
  12. J

    Why can no one explain Power Series and Functions clearly

    Hello, Im currently in a Calc II class with unfortunately a bad professor (score of 2 on RateMyProfessor), so I often have to resort to outside sources to learn. Our class is currently on Sequences and Series which has been fine up until we hit the topic of relating Power Series and Functions...
  13. S

    Help with infinite sequences and series

    I tried the comparison test for one B but not sure if I am right. Think it could also be a ratio test because of the variable exponent. I'm lost totally lost on number one A. Also, I have the answer for the first part of three but don't know how to do the second part of it by comparing. Thanks
  14. A

    MHB Investigate Convergence of tanx Series: Find Common Ratio & Sum to Infinity

    Hi, Please help me with this question: Investigate the convergence of the sequence tanx;tan2x;tan3x;...;tannx for xE(-90;90 degrees). Steps to follow: Find common ratio. Draw the graph. For which values will x converge. Determine sum to infinity. I did try to solve, but file type too...
  15. V

    Can someone check my explanation of sequences and series?

    It's been a while since I've dealt with sequences and series. Here is my explanation of sequences and series and let me know if I am right or wrong. A sequence is just a list of numbers. By convention, we use the letter ##a## for sequences and they are written in a form like so...
  16. C

    Solve Sequences & Series Homework: Find k for (6+3n)^-7

    Homework Statement So, I actually have a bunch of these problems and I cannot do any of them. I don't think I'm really understanding it. Here is the question: (one of them) The way I wrote them, a_n means a sub n For each sequence a_n find a number k such that n^k a_n has a finite non-zero...
  17. B

    The only method to learn sequences and series

    I remember when I took Calculus B in college. I had never learned any math by reverse engineering before, but when I got to sequences and series, the only way for me to learn how to do it was to reverse engineer it. I had to look up the answer in the back of the textbook, and then work...
  18. G

    Sequences and Series of Functions Question (Rudin Chapter 7)

    Homework Statement The problem is Exercise 8 from Chapter 7 of Rudin. It can be seen here: http://grab.by/mGxY Homework Equations The Attempt at a Solution It seems quite obvious to see that because \sum\left|c_n\right| converges, f(x) will converge uniformly. However...
  19. T

    Sequences and Series Problem

    Homework Statement The sum of the first 9 terms of an arithmetic series is 216. The first, third and seventh terms of the series form the first three terms of a Geometric pattern. Find the first term and the common difference of the arithmetic pattern. Homework Equations The Attempt at a...
  20. M

    Sequences and Series exercises text

    Hello, In Calculus 2, sequences and series are introduced and do I have to say that most of the examples are trivial and even the exercises are either trivial or those that require experience. I hope someone can suggest a book where one can learn solving not-so-obvious series problems that...
  21. V

    Sequences and series problem

    Homework Statement If P r=(n-r)(n-r+1)(n-r+2)...(n-r+p-1) Qr= r(r+1)(r+2)...(r+q-1) Find P1Q1+P2Q2+... +Pn-1Qn-1 Homework Equations The Attempt at a Solution I tried to bring the general term in...
  22. C

    Infinite sequences and series help

    Hi I don't understand the logic in the picture i added. They say that "that sum of the series = the limit of the sequence" The limit is 2/3 BUT the sum, Ʃ, must be 2*1/(3*1+5) + (2*2/(2*3+5) + 2*3/(2*3+5) ...+ Which is obviously much larger than 2/3 if all the terms are added together?? it's...
  23. U

    Sequences and series problem

    Homework Statement Read this passage and then answer the questions that follow We know that, if a_1,a_2,...,a_n are in Harmonic Progression, then \frac{1}{a_1},\frac{1}{a_2}...,\frac{1}{a_n}, are in Arithmetic Progression and vice versa. If a_1,a_2,...,a_n are in Arithmetic Progression with...
  24. B

    Solving Sequences & Series: Limits & Sums

    Hello, I am curious to know that if we take some seqence, a_n, and take the limit as the the terms of the sequence goes to infinity, will the sequence head towards the same value that the the sum of the infinite amount of terms added together? (I hope I worded that properly...)
  25. L

    Sequences and series help .

    Sequences and series help... [b]1. Homework Statement 3+3a+3a^2+...∞ is = to 45/8 where a>0,then a is...? [b]3. The Attempt at a Solution since it is a g.p so using S=(a(rn-1))/(r-1) for r>1 ive all the values except for "n"..can someone help...:/
  26. L

    Infinite sequences and series - conv or div - sigma(e^(1/n)/n)

    Homework Statement Determine whether the series converges or diverges Sum from n=1 to infinity ((e^(1/n))/n) Homework Equations I am trying to use the limit comparison test to prove it. The Attempt at a Solution an = (e^(1/n))/n bn = e/n an/bn = e^(1/n)/e lim n->...
  27. T

    Sequences and Series Problem. Help Pleeaassee

    Sequences and Series Problem. Help! Pleeaassee! Homework Statement I've attached the problem and my work. I'm supposed to express the sum as a fraction of numbers in lowest terms. The original statement was: 2/(1*2*3) + 2/(2*3*4) + 2/(3*4*5) + ... + 2/(100*101*102) and the answer is 2575/5151...
  28. D

    Sum of Products of All Pairs of Natural Numbers

    Homework Statement Find the sum of the products of every pair of the first 'n' natural numbers. Homework Equations sigma n^2 = n(n+1)(2n+1)/6 The Attempt at a Solution S=1.2 + 1.3 + 1.4 ...+ 2.3 + 2.4 ...n-1(n) i can't figure out how to proceed ..
  29. O

    Geometric Sequences and Series

    Homework Statement Q.: Show that if log a, log b and log c are three consecutive terms of an arithmetic sequence, then a, b and c are in geomtric sequence. Homework Equations Un = a + (n - 1)d and Sn = \frac{a(r^n - 1)}{r - 1} The Attempt at a Solution Attempt: Consider...
  30. O

    Geometric Sequences and Series

    Homework Statement Q.: The sum of the first five terms of a geometric series is 5 and the sum of the next five terms is 1215. Find the common ratio of this series. Homework Equations Sn = \frac{a(r^n - 1)}{r - 1} The Attempt at a Solution a + ar + ar^2 + ar^3 + ar^4 = 5 ar^5 +...
  31. N

    Sequences and series question

    Homework Statement I wrote a test and the question was something like this 2, 4, 6... 108 It said... " Find tn" Does this just mean find any term number that isn't given? I just plugged in t5 for the arithamtic logic and solved.. don't know if it was right, does anyone know...
  32. O

    Arithmetic Sequences and Series

    Homework Statement Just a quick question I was looking to have cross checked… Q. Find un, the nth term of sequence -5, 0, 5, 10,… Homework Equations un = a + (n-1)d The Attempt at a Solution -5 + (n-1)5 -5 + 5n - 5 5n-10 The answer in the book...
  33. P

    Population Growth in Country: Calculating +1 & +4 Year Populations

    The population of a country is 15.2 million and is growing at a steady rate of 2.7% annually. a) What was the population one year ago b) What was the population four years ago So I did this: Un+1 = Un-1 * ((n-1)*0.973) Doing so, I got for the first answer 14,789,600 but the book got...
  34. J

    General formula for finding the sum of sequences and series?

    I know that there are particular formulas for finding the geometric/arithmetic/ and recursive sequences or series with \Sigma. But is there a general formula for finding the sum for all three types? For example, what if I was asked to find a sum of a particular finite sequence but I don't know...
  35. C

    Sequences and series question

    ! Sequences and series "limit" question, is my solution correct? Homework Statement [PLAIN]http://img233.imageshack.us/img233/7195/sands2010q1.gif [Broken] Homework Equations The Attempt at a Solution Solution posted in image above, want to know if its correct
  36. Char. Limit

    Comparing Sets of Convergent Sequences and Series

    So I had this question in PF chat, but I decided this would be a better place for it. Say I have two sets, S and S'. S is the set of all convergent sequences. S' is the set of all convergent series...es. Is S larger than S', and if so, how much larger?
  37. Char. Limit

    Solving Sequences and Series Homework

    Homework Statement So I was helping my roommate with his homework, and it has the following problem: Homework Equations The Attempt at a Solution We tried a Fibonnaci-type sequence, but that really didn't work. And we don't know any other types of sequences. Should I try some...
  38. P

    Solving Sequences and Series Problems - Struggling!

    Struggling with this topic! :( got a couple of questions. Homework Statement 1) Determine the value of the improper integral when using the integral test to show that \sumk/(e^k/5) is convergant given answers are a)50/e b)-1/(5e^1/5) c)5 d)5e e)1/50e 2) determine whether \sum...
  39. I

    Solve Series P: Sum of First n Terms & Arithmetic Progression

    CAN U ALL HELP ME TO SOLVE THIS QUESTION? I don't know how to start... The sum of the first 2n terms of a series P is 20n-4n2. Find in terms of n, the sum of the first n terms of this series. Show that the series is an arithmetic series.
  40. S

    Real Analysis: Sequences and Series

    Suppose that ak is a decreasing sequence and (ak) approaches 0. Prove that for every k in the natural numbers, ak is greater than or equal to 0. I was thinking I should assume the sequence is bounded below by 0 and do a proof by contradiction. Any suggestions?
  41. F

    Sequences and Series of Functions

    Homework Statement Let sum of a sub k be an absolutely convergent series. a. Let f be the function defined by f(x) = sum of (a sub k) * sin(kx). Prove that: the integral from 0 to pi/2 of f = sum of (a2k-1 + a4k-2)/(2k-1) Homework Equations I already showed that f(x) converges...
  42. A

    Sequences & Series: Limit to Determine Convergence/Divergence

    How do you know if a sequence converges or diverges based on taking the limit? here's an example f:= 3^n/n^3; if i take the limit the sequence goes to infinity. does it diverge becuase the limit is not zero or can the limit be something other than zero and it still converge?
  43. L

    Solve Sequences and Series: Total Distance 480m

    Any guidance or worked solutions would be appreciated Homework Statement For a potato race, a straight line is marked on the ground from a point A, and points B,C,D,... are marked on the line so that AB = BC = CD = ... = 2 metres. A potato is placed at each of the points B,C,D,... A...
  44. T

    Sequences and series help / recurrence relation

    Homework Statement A sequence of terms U_{k} is defined by K \geq by the recurrence relation U_{k+2} = U_{k+1} - pU_{k} where P is a constant Given that U_{1} =2 and U_{2} = 4 a) find an expression in terms of p for U_{3} b) hence find an expression in terms of p for...
  45. K

    Best way to approach sequences and series

    I get the impression that unlike solving derivatives and integrals, sequences and series do not have a lot of...should I say...find-the-equation-and-solve-your-way element -- sorry if that comes out wrong. Maybe it seems to be less "rote math" and because of this, I'm having a hard time trying...
  46. W

    Calc II and difficulty of sequences and series

    Ok so I am in Calculus II this summer and its pretty easy so far. However, I have heard the hardest part about Calc II is series and seqence. Why so? And what can I do to make it easier on myself? What was your expierence with sequence and series. Thanks in advance.
  47. D

    Sequences and Series Help

    This was the extra credit question on a quiz I had today, I am very anxious to find out the answer. 1. Homework Statement Find the apparent Nth term of the sequence 2,-5,10,-17 ... n 2. Homework Equations Not sure really on this an = ... 3. The Attempt at a Solution...
  48. P

    Sequences and series

    A={xεR:X^11+2X^5<2} let a=supA By choosing a suitable sequence of elements of belonging to A and which tends to a as n->inf, or otherwise, show that a^11+2a^5=<2.Choose another sequence this time of all real numbers not belonging to A to show that a^11+2a^5>=2 and hence show that a^11+2a^5=2,so...
  49. O

    Sequences and series - try again

    Sequences and series - try again :) Hi, I'm going to try to post this question again, hopefully it is more clear this time. I'm not sure how to approach this question, or really, what this question is asking me! Homework Statement The k-th term of a series, Sk = a*[(1-(r^k))/(1-r)], is...
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