Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

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Better understanding of a series

Let $B=\{x\in\mathbb{R}^N:\ |x|<1\}$ with $N\geq 2$ and $x_n$ a countable dense set in $B$. Consider the function $$u(x)=\sum_{i=1}^\infty\frac{1}{2^{i}|x-x_i|^{1/2}},\ \forall\ x\in B$$ By using, ...
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34 views

As resolved the identity of this series?

Use the identity $\cos((k-\frac{1}{2})x) - \cos((k+\frac{1}{2})x) = 2\sin kx \sin \frac{x}{2}$ to show that $S_n:=\sum_{k=1}^{n}\sin kx=\frac{1}{2\sin \frac{x}{2}}(\cos\frac{x}{2}-\cos((nx+\frac{x}{...
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78 views

Reformulate summations of undefined length

I have the following equation that is not "writable" as its length depends on the variable t. So, I'd like to reformulate it. $N(t)=\sum_{i=1}^{t}\sum_{j=1}^{i}\...
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25 views

Reformulate some summations

Is there a way to reformulate the following equations (leaving N(...) alone on the left-hand side): $$N(t)=\sum_{a=1}^{t}\sum_{b=1}^{a}\sum_{c=1}^{b}\sum_{d=1}^{c}...
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2answers
554 views

A basic question on subsequential limits

Suppose we have a sequence $\{x_n\}$. Consider the set $S$ of subsequential limits of $\{x_n\}$. Suppose, for any given $\epsilon$ I do the following experiment : For each subsequential limit $x^{(k)}$...
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1answer
170 views

Finding a generating function for a pattern

I was working on this projecteuler.com problem, and I was very interested by the premise. Essentially, given n terms, find an ...
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2answers
109 views

A result about infinite series: How to prove this?

Let $\{a_n\}$ be a sequence of positive real numbers such that, for some $N \geq 1$, some $s>1$, and some $M>0$, we have $$ \frac{a_{n+1}}{a_n} = 1 - \frac{A}{n} + \frac{f(n)}{n^s} $$ for all $...
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71 views

Nonnegative series converges implies terms decay exponentially?

Let $\{a_n\}$ be a sequence with nonnegative terms ($a_n\geq 0$). If $\sum_{n=0}^\infty a_n < \infty$, does this imply that there exists $\alpha<1$ such that $a_n \leq \alpha^n$ for all but ...
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190 views

Infimum of a set of a sequence of numbers

Consider the sequence $\{ y_{n} \}$ of real numbers such that $\sup\{\left | y_{n} \right |:n\in \mathbb{N}\}=4$. Find $\inf \left\{\frac{\left | y_{n} \right |}{n}:n\in \mathbb{N} \right\}$ Since $4=...
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1answer
211 views

Series with an exponential term containing a sum

can anyone help me with the convergence of this series $\sum_{x=1}^\infty \exp(-c\sum_{j=1}^x j^\beta)$ for $-1<\beta<0$? Which convergence criterion can be useful here? Thanks
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102 views

Outer Lebesgue measure - countable sub-additivity

Denote $$ m^*(\Omega) := \inf \left \{ \sum_{k=0}^\infty vol(B_k) : (B_k)_{k=0}^\infty \text{ covers } \Omega \right \} $$ where $\Omega \subseteq \mathbb R^n$ and the $B_k$ are open boxes of the form ...
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2k views

Find the period of sequence.

A sequence is such that its terms are generated by the formula: $$r_i =(ar_{i-1}+b\pmod m)$$ where $a,b,m,r_0$ are given. find the period,that is the number of terms that are repeated. For example, ...
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1answer
44 views

alternating between high values and low values

I want to generate $y(x)$ such that $x=1$ to $100$ $y(x)=1,2,3,10,20,30,7,8,9,70,80,90,10,11,12$ the idea is to come up with an equation where $y(x)$ alternate between high values and low values.... ...
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30 views

Data estimation based on progression

Given a data-set $x$ and $y$. x | y ------------------ 153,000 | 0.058848 332,641 | 0.36352 506,629 | 0.53 If $x$ being the number of database records ...
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1answer
58 views

exercise oriented to define $a^{x}$ for $a, x$ real numbers

This exercise is oriented to define $a^{x}$ for $a, x$ real numbers Let $s_{n}$, $r_{n}$ two sequences of rational numbers that converges to $x$, then there exist a real number $L$ such that $a^{s_{n}...
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379 views

Calculation the variance of the forecast error?

Hi there stuck on the following: Consider the model: $$y_{t}=(1+a)y_{t-1}-(a)y_{t-2}+\epsilon_{t}$$ where $\epsilon_{t}$ is a white noise problem: 1) Transform $y_t$ into some other series $w_t$ ...
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1answer
202 views

Recurrence equation with upper and lower boundary condition

A very natural set up for recurrence equations is the following: $$ s(0) = 0 $$ $$ s(k) = A \ s(k-1) + B $$ $$ s(M) = A \ s(M-1), $$ where $0 \le A,B \le 1$ and $0 < k < M$. We can omit the ...
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2answers
58 views

Finding the explicit formula of the inhomogeneous difference

I have the following inhomogeneous difference equation: $$z_{t+1}-z_{t}=1$$ Solving as a homogeneous equation, I get $z_{t}=A(1)^{t}$; however, when solving for the inhomogeneous case, can I try ...
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1answer
42 views

Upper Bounds of Two Interdependant Recursive Sequences

For a pair of real numbers $\alpha$ and $\beta$, I need to prove that for the sequences $a_n = (-\alpha)a_{n-1} +b_{n-1}$ $b_n = (-\beta)a_{n-1}$ an upper bound exists with a form similar ...
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58 views

Function of series

Let us consider the following function $$F(x) = \sum_i \frac{a_i}{x - y_i}$$ Is it possible to simplify $F(x)$ so that the repeated sum for each value of $x$ can be avoided ?
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293 views

Calculating probabilities in genetic sequences

I am working with certain recurring sequences in genetics and try to calculate certain probabilities: Let for instance $$\langle g_i\rangle :=\{1,1,1,6,1,1,1,6,...,1,1,1,6\}$$ and $$\langle h_i\...
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2answers
57 views

Positive limit of sequence vs. positive terms

Let $\{x_m\}$ be a sequence in $E_1$ that converges to $L \in E_1$. a. prove that if $L>0$ and there exists $n \in N$ such that for all $m >n$ holds that $x_m > 0$ b. True or false? If for ...
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75 views

question involving the limit of geometric series

We have that $a\in\mathbb{R}$ is s.t. $a>1$. I was wondering what happens to the following as $N\rightarrow\infty$. $$\frac{a^{2N-1}+a^{2N-3}+\cdots+a}{a^{2N}+a^{2N-2}+\cdots+1}x_1 + \frac{a^{2N}}...
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47 views

Interesting D.E turns into Geometric Limit and Cyclic Group.

Me and a friend Observed some very odd behavior of a matrix when trying to solve an ODE Consider first the 3x3 system where a,b are parameters in the Reals and fine the general solution. $A=\begin{...
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1answer
325 views

Showing convergence for certain values of P

Determine for which values of p the following series converges: $$\sum_{n=2}^{\infty} (-1)^{n-1} \frac{(\ln n)^P}{n}$$ So far, just from looking at various values for $P$, it seems to be that any $P$...
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77 views

Limit points of a sequence contained in $S^1$

Let $\theta\in (0,2\pi)$ be a real number such that $\displaystyle\frac{\theta}{\pi}\notin\mathbb{Q}$. We define $z:=\cos(\theta)+i\sin(\theta)\in S^1\subseteq\mathbb{C}$ and let $\{z_n\}_{n\in\mathbb{...
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467 views

Iterate over combinations ordered by sum

I have a sorted list of a large number of primes. I want to iterate over combinations of fixed size $n$ in increasing order of their sum. Naturally the standard approach for $n=4$: $$s_0 = \sum(A, B,...
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599 views

Show a series of functions has a continuous sum.

For every $x \in \mathbb{R}$ define $$I(x) = \begin{cases} 0 & \text{if} & x \leq 0,\\ 1 & \text{if} & x > 0 \end{cases}$$ Suppose that $(x_n)$ is a sequence of points in $(a, b)$ ...
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104 views

logistic difference equation problem

Consider logistic difference equation $${{x}_{n+1}}-r{{x}_{n}}\left( 1-{{x}_{n}} \right)=f\left( x \right),\ \ 0\le {{x}_{n}}\le 1\ \ \ \ \ \ \left( 1 \right)$$ 1.Show hat expression $$f\left( f\left( ...
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1answer
127 views

Finding the lowest sum of the sequence possible given some conditions

Say we have an infinite sequence of natural numbers $A$ such that no $k$ subsequences can be found adjacent such that the average of the elements in any subsequence is equal for all $k$ subsequences. ...
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220 views

Evaluating an infinite sum

I want to find the value of this sum. $$\left(\frac{b}{1-ab}\right)\sum\limits_{n=0}^{\infty}\left(c+ (n+1)I\right)^d\left(\frac{ab}{1-ab}\right)^n $$ My thoughts: Writing out the first terms $$S = ...
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173 views

Convergence of (infinite) potence tower?

Let $(a_n)_{n\in\mathbb{N}}$ be a series in $\mathbb{C}$ or $\mathbb{R}$. Which contraints must $(a_n)$ match to make $b_n := a_1^{a_2^{...^{a_n}}}$ converge for $n\rightarrow\infty$? For constant ...
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1k views

Cauchy Sequences bounded away from zero.

Is there a positive real number which can be written as a Cauchy sequence, such that this Cauchy sequence is bounded away from zero and also this sequence contains infinite number of positive and ...
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1answer
62 views

Inductive Proof to Analyze an Arithmetic Series

I have a particular number series as follows:-7, 3, 5, 13 and 27. Please assume as correct that -7 represent 7 points in 0D, 3 represents 3 points in 1D, 5 points for 2D, 13 points for 3D, and 27 ...
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89 views

Mathematical Analysis-Cluster points-Bolzano Weistrass THM

I am stuck in my one of the homework problems, the question is like the following: Let $(x_n)$ be a bounded sequence, and let $c$ be the greatest cluster point of $(x_n)$: (a) Prove that for every $...
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3k views

Algorithm for conversion of seconds into date/time?

What mathematical algorithm would I use to convert seconds since 1/1/1970 into a usable date/time format (time can be left off)? This includes leap years with extra days and any other additions. I'm ...
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1answer
41 views

Is there a specific terminology for numbers which are nontrivial multiples of triangular numbers?

(Note: Please see this new question for the motivation.) A number $T$ is said to be triangular if it could be written in the form $$T=\frac{n(n+1)}{2},$$ where $n$ is a positive integer. Here is my ...
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1answer
142 views

Find a formula of sequence: $\frac{-1}{2}, 0, \frac{1}{10}, 0, \frac{-1}{26}, 0, \frac{1}{50}, 0, \frac{-1}{82}, 0, \frac{1}{122}, 0, \dots$

I'm working on a discrete math homework that finding a formula for the following sequence: $$\frac{-1}{2}, 0, \frac{1}{10}, 0, \frac{-1}{26}, 0, \frac{1}{50}, 0, \frac{-1}{82}, 0, \frac{1}{122}, 0, \...
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2answers
58 views

Uniqueness of coefficients in infinite sin series $a(x) = \sum\limits_{n=1}^{\infty} a_{i} \sin(n x) =0 $

if we can say $$a(x) = \sum\limits_{n=1}^{\infty} a_{n} \sin(n x) =0, \forall x$$ does this generally imply that all constant coefficients $a_i$ should be zero, or can I construct any number of ...
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3answers
92 views

Solving a recurrence relation: can't figure out how to convert from summation

I am really struggling to solve this recurrence. $$ T(n) = T(\sqrt{n}) + n. $$ I am asked to give asymptotic upper and lower bounds for $T(n)$. I am free to use any method to arrive at my answer, ...
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1answer
55 views

Find the recurrence formula!

I have a sequence defined by recursion as follows: $$\begin{cases}x_0=a\\ x_{n+1}=x_n\cdot B^{x_n} \end{cases}$$ where $a,B$ are fix natural numbers. Does anyone know how to find a recurrence formula ...
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1answer
53 views

Finding the common difference and hence, the sum of an A.P

Find the sum to $25$ terms of an A.P with the first four terms as $1, \log_yx, \log_zy,-15\log_x z$. My attempt: I started out with, $2\log_yx = 1+\log_zy$ and, $2\log_zy = \log_yx -15\log_xz$ ...
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1answer
61 views

Uniform convergence of $ U_n(x) = \sum_{n=0}^{+ \infty} (-1)^n \ln ( 1 + \frac{x}{1+ nx} ) $.

We consider the series of functions: $$U_n(x) = \sum_{k=0}^{n} (-1)^k \ln \left( 1 + \frac{x}{1+ kx} \right) ,~ x \geq0.$$ Prove that $U_n$ is convergent. Study the uniform ...
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1answer
69 views

Harmonic series (Maths and Music)

I am a high school student trying to apply calculus to music (harmonic series). I am just wondering, how can I collect data from any online music app (with music tones that form harmonic series - ...
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1answer
37 views

Prove that the function can be continued into a larger domain

Prove that the function $f(z)=\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}\frac{z^n}{n}$ can be continued into a larger domain by means of the series $$\ln2-\frac{1-z}{2}-\frac{(1-z)^2}{2\cdot 2^2}-\...
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1answer
61 views

A conjecture on bounded complex partial sums

A friend of mine has made the following conjecture, but we don't know how to prove it. Let $(a_n)_{n\in \mathbb {N}}$ be a strictly increasing sequence of natural numbers. Suppose that for every ...
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1answer
139 views

When is the product of limsups equal to the limsup of the products?

Let $\left\{x_{n}\right\}$ be a sequence of real numbers where $x_{n} > 0$ for all $n \in \mathbb{N}$. Given that \begin{equation*} x = \limsup_{n \rightarrow \infty} x_{n}^{1/n} = \limsup_{n \...
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2answers
41 views

Technique for using direct and limit comparison tests?

Suppose you have an infinite series $$\sum_{n=1}^{\infty} \frac{1}{n^2 + 9}$$ Using the direct comparison test, the sequence can obviously be compared to $\sum_{n=1}^\infty\frac{1}{n^2}$ since it is ...
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1answer
66 views

For which value of $x \in \mathbb{R} $ does $\sum\limits_{n=1}^\infty x^{n\log n}$ converge?

For which value of $x \in \mathbb{R} $ does the following series converge: $$\sum_{n=1}^\infty x^ {n\log n}.$$ The series of the absolute values is $$ \sum_{n=0}^\infty |x|^ {n\log n},$$ and ...
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1answer
79 views

How to solve the system $x_{t+1}=-x_t-2y_t+3t-2$, $y_{t+1}=-2x_t+2y_t+t+1$

I have the following system of recurrence equations: $$x_{t+1}=-x_t-2y_t+3t-2\qquad y_{t+1}=-2x_t+2y_t+t+1$$ I write this in matrix-vector form: $$r_{t}=Ar_{t-1}+b\cdot (t-1)+c$$ I repeatedly ...