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.
1. Finding the limit of the sequence:
{ an } = 5n^(2) / (n^(2) + 2)
Homework Equations
3. what i did was :
lim as (n -> Infinity) of function [5n^(2) / (n^(2) + 2)]
Then factored out the constant:
5{lim as (n -> Infinity) of function [n^(2) / (n^(2) + 2)]}...
Hi I am playing around with recursive definitions of Lucas and Fibonacci sequences:
I came across a relationship
L0 + L1 + L2 + L3 ... Ln = sum(i = 0, n) Li = Ln+2 -1;
Sorry for the horrible notation, but could anyone provide a counter example using an inductive approach? I get the...
Homework Statement
Determine the convergence, both pointwise and uniform on [0,1] for the following sequences :
(i) ##s_n(x) = n^2x^2(1 - cos(\frac{1}{nx})), x≠0; s_n(0) = 0##
(ii) ##s_n(x) = \frac{nx}{x+n}##
(iii) ##s_n(x) = nsin(\frac{x}{n})##
Homework Equations
##s_n(x) →...
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...
Homework Statement
what is the general formula for the sequence (1/1*3+1/3*6+1/6*10+1/10*15...)
Homework Equations
i used the equation n/mn+1 but am not able to use it for this sequence
The Attempt at a Solution
I found the sequence of the denominators which is (1/2)n^2+(1/2)n...
I have to prove that the cardinality of the set of infinite sequences of real numbers is equal to the cardinality of the set of real numbers. So:
A := |\mathbb{R}^\mathbb{N}|=|\mathbb{R}| =: B
My plan was to define 2 injective maps, 1 from A to B, and 1 from B to A.
B <= A is trivial, just...
I'm trying to find a sequence that has subsequences that converge to every integer. The question before that was the same but just for the positive integers, for which i gave {1,1,2,1,2,3...} but I'm struggling to include the negatives. Thanks
Homework Statement
For an increasing sequence of numbers, how many other sequences could this be the average sequence of.
Homework Equations
Where the average sequence, a[i] = 0.5( s[i] + s[i+1] )
The Attempt at a Solution
If there's n terms in the original sequence.
The number of...
Hi,
True or False: Every infinite sequence of natural numbers, who's terms are randomly ordered, must contain every possible subsequence of any length, including infinity.
For example, does the infinite and random sequence \small M of natural numbers require that the subsequence {59,1,6}...
After losing marks in an exam due to significant figures, I have decided to clear up all my doubts about this concept. But since my teacher hasn't been very helpful, I've decided to post my question here.
I understand the rules for significant figures in both single-step...
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...
In this link:
http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw3sum06.pdf
For number 8.4...
Why don't we just say...
|s_n||t_n| < \frac{\epsilon}{M} M = \epsilon?
Thanks in advance
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...
Homework Statement
I'm having trouble with these here.. it's been a while since I've done sequences and I can't seem to make this work with Standard Limits equations.
Clearly the answer given by Wolfram solver is there after the = but i'd like to know the reasoning behind it.
Anyone that...
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...
Homework Statement
Find the sum of the sequence:
2, -2/3, 2/9, -2/27, 2/81, . . .
Homework Equations
The Attempt at a Solution
I can see that the number is multiplied by -1/3, but I'm unsure of how to find the sum.
Any pointers?
The effect of changing values in sequences??
I have been given a maths assignment and have been given equations \(u_{n+1}=2u_{n}+2\) and asked what is the effect if the value \(u_{0}\) is changed? I used multiple values both positive and negative and have only noticed taht when it is a high...
Homework Statement
Let an be a bounded sequence and bn such that
the limit bn as n→∞ is b and
0<bn ≤ 1/2 (bn-1)
Prove that if:
an+1 ≥ an - bn,
then
lim an
n→∞
exists.
Homework Equations
The Attempt at a Solution
as 0<bn ≤ 1/2 (bn-1) the sequence bn is...
Homework Statement
Let an be a bounded sequence and bn such that
the limit bn as n→∞ is b and
0<bn ≤ 1/2 (bn-1)
Prove that if:
an+1 ≥ an - bn,
then
lim an
n→∞
Homework Equations
The Attempt at a Solution
no clue :(
Homework Statement
Let f be a 2π-periodic function (can be any periodic really, not only 2π), and let g be a smooth function. Then
lim_{n\rightarrow∞}\int^{B}_{A} f(nx)g(x) converges to \frac{1}{2π}\int^{2π}_{0}f(x)
The Attempt at a Solution
So far, I've come up with somewhat of...
Homework Statement
Prove the following theorem, originally due to Cauchy. Suppose that (a_{n})\rightarrow a. Then the sequence (b_{n}) defined by b_{n}=\frac{(a_{1}+a_{2}+...+a_{n})}{n} is convergent and (b_{n})\rightarrow a.
Homework Equations
A sequence (a_{n}) has the Cauchy property...
Let x_n and y_n be two convergent sequences with different limits. Show that the set {x_n : n€N} n {y_n : n€N} is finite.
Attempt: by definition, for each £>0 there exists an N such that |x_n - x|<£ and similarly |y_n - y|<£ holds for every n with n>N. Take £=(x-y)/3 and assume that x_n and...
Hello everyone!
Let $a_n$ and $b_n$ be two sequences such that $a_n \leq b_n$ for all $n$. Let $A_n = \sup \{a_m \; | \; m \geq n\}$ and $B_n = \sup \{b_m \; | \; m \geq n\}$.
I want to prove that $A_n\leq B_n$. I attempted a proof by contradiction:
Assume $A_n > B_n$ for some $n$.
If $A_n =...
I have a linear recurrence sequence and am having a problem understanding what to do when the ratio does not seem to be the same between each of the terms, so
Terms;
4, 1.4, 2.44, 2.024... (n = 1,2,3...)
How do I find a the ratio of these terms, and if there is none, please advise how I...
cn= (4n)/(n+4n^(1/n))
When i set it up i think i should use l'hopital but I am confused what to do with the 4n^(1/n) term.
an=(7^(2n))/(n!)
I know this is a geometric sequence and top and bottom increase initially then tend to 0, but I am lost on how to show the work. should i expand...
Homework Statement
Assume that \{ a_n\}\rightarrow 0 . Use the definition of limit to prove that \{ a_n^2\} \rightarrow 0.
Homework Equations
Definition of limit. For all ε>0 there exists N s.t. n>N implies |a_n - L|<ε.
The Attempt at a Solution
I know why this is true... if the sequence...
Homework Statement
I have to proof that the sequence (2^n +n^2)/(3^n + 5n^4) converges en calculate its limit using the sqeeuze theorem.
Homework Equations
(2^n +n^2)/(3^n + 5n^4)
http://www.proofwiki.org/wiki/Squeeze_Theorem#Sequences
Theorem 1: Let p\in2N en x\inR with |x|< 1. Then the...
Homework Statement
Let (xn)n\inℕ and (yn)n\inℕ be Cauchy sequences of real numbers.
Show, without using the Cauchy Criterion, that if zn=xn+yn, then (zn)n\inℕ is a Cauchy sequence of real numbers.
Homework Equations
The Attempt at a Solution
Here's my attempt at a proof:
Let...
Hi,
I am wondering how one would go about an ε, N proof for a recursively defined sequence. Can anyone direct me to some reading or would like to provide insights of their own? This isn't for a homework problem... just general curiosity which I could not satisfy via search!
Thank you.
Homework Statement
I want to prove that if a sequence a[n] is cauchy then a[n] is a converging sequence
Homework Equations
What I know is:
a[n] is bounded
any subsequence is bounded
there exists a monotone subsequence
all monotone bounded sequences converge
there exists a...
I am not getting anywhere with this problem.
Prove the Schwarz's and the triangle inequalities for infinite sequences:
If
$$
\sum_{n = -\infty}^{\infty}|a_n|^2 < \infty\quad\text{and}\quad
\sum_{n = -\infty}^{\infty}|b_n|^2 < \infty
$$
then
$\displaystyle\left(\sum_{n = -\infty}^{\infty}|a_n +...
Sorry for the rather vague title!
Homework Statement
Given:
Two Banach spaces A and B, and a linear map T: A\rightarrow B
The sequences (x^n_i) in A. For each fixed n, (x^n_i) \rightarrow 0 for i \rightarrow \infty.
The sequences (Tx^n_i) in B. For each fixed n, (Tx^n_i) \rightarrow y_n...
I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from
http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded
I don't understand that how we can take m=N_{1} while m>N_{1} ?
In fact i mean that in a metric space (A,d) can we say that...
If A and B are vector spaces over ℝ or ℂ show that a sequence (a_n, b_n) in A×B converges to (a,b) in A×B only if a_n converges to a in A and b_n converges to b in B as n tends to infinity.
To me this statement sounds pretty intuitive but I have been having trouble actually proving it...
Homework Statement
\stackrel{lim}{n\rightarrow \infty} (-1)^n \frac{n}{n + 1}
Homework Equations
The Attempt at a Solution
The answer is that the limit oscillates between -1 and 1, but I was wondering if there was an analytic was of showing this.
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...)
Homework Statement
Prove that the given sequence diverges to infinity.
{an} = (-n^4+n^3+n)/(2n+7)
Homework Equations
Diverges definition
The Attempt at a Solution
So far I have:
Let M>0 and let N= something.
I'm having a hard time figuring out what N should equal for the...
Homework Statement
Determine whether the given limit exists and find their values. Give clear explanations using limit properties.
Homework Equations
lim n--->∞ (n^2)/n!
The Attempt at a Solution
I know that the limit is 0, but I don't know how to show it in detailed steps...
can anyone show me how to do this question ? thanks ...
express (1+x^2)/((1+x)(1+2x)) in partial fraction. (this step i know the solution )
hence,find the constant term in the expansion if (1+x^2)/(-3x(1+x)(1+2x)) in ascending power of x .( then this one don't know ,please help me ) thanks ...
I was attempting to find a counterexample to the problem below. I think I may have, but was ultimately left with more questions than answers.
Consider the space, L, of all bounded sequences with the metric \rho_1
\displaystyle \rho_1(x,y)=\sum\limits_{t=1}^{\infty}2^{-t}|x_t-y_t|
Show that a...
Hi guys, I'm doing some exercises in which given a recursive sequence and its first term, I have to find the general formula/term. I am stuck in two and I would like some help. Thanks in advance. Now, the sequences:
1) a1=1, an+1= an + ((-1)^(n+1))n^2
So, the first terms are: a2=2...
Can someone guide me toward using my TI-89 Titanium calculator for sequences?
I would like to be able to PUT IN a sequence of numbers and have it GIVE ME the formula. Not vice versa please. Thanks.
Hello.
Having already learned about infinite series and sequences in my calculus class, I'm quite interested in them and especially in learning more about them. If any of you have in mind any good books on the subject which you can recommend to me, it will be very much appreciated...
Def: A low discrepancy sequence is a uniformly distributed sequence with minimal discrepancy, O(logN/N).
Question: Let <x> denote the fractal part of an irrational number x. Let (<x_n>) be an arbitrary low discrepancy sequence. Is it always true that :
\lim_{n \to +\infty}|<x_n - x_{n-1}> -...
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...:/
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->...
Hi everyone,
I'm doing an investigation of markov properties and in an example I have made the following transition matrix:
http://img152.imageshack.us/img152/1584/matrixki.png
If all the probabilities were above zero, finding the total number of possible 4-state sequences (i.e. ACBA, BACB...
Homework Statement
Suppose that a mathematically inclined child plays with a basket containing an infinite subset of integers (with some repetitions). If an integer k is present in the basket then there are initially |k| copies of it. The child pulls out the integers from the basket at...
Hi, I have the following problem and have done the first two questions, but I don't know how to solve the last two. Thanks for any help you can give me!
Homework Statement
Let a_{n}\rightarrow a, b_{n}\rightarrow b be convergent sequences in \Re. Prove, or give a counterexample to, the...
Homework Statement
Find the first term in this geometric sequence that exceeds 500.
2, 4, 8, 16, ...
Homework Equations
Un = arn-1
The Attempt at a Solution
a = 2, r = 2
Un = 2 x 2n-1 > 500
2 x (2n)(2-1) > 500
log22 x log22n + log22-1 > log2500
1 x n + (-1) > log2500
n - 1 >...