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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7
votes
2answers
217 views

Factorizing the Sum of Two Fibonacci numbers

The Fibonacci and Lucas numbers are defined for all integers $n$ by the recurrence relations $$F_n=F_{n-1}+F_{n-2}\text{ where }F_1=1\text{ and }F_2=1,$$ $$L_n=L_{n-1}+L_{n-2}\text{ where }L_1=1\text{ ...
7
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1answer
158 views

Does there exist infinite words using the alphabet $\{A,B,C,D\}$ that avoids patterns $XX,\ XAX,\ XBX,\ XCX,\ XDX$?

Another form of this question is: Does there exist a gap-1 square-free infinite word using the alphabet {A,B,C,D}? Normally square-free in this context means that there are no sub-words twice in a ...
6
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0answers
111 views

Generalized limits

$\DeclareMathOperator{\Lim}{Lim}$ $\DeclareMathOperator{\dom}{dom}$ $\DeclareMathOperator{\shift}{\sigma}$ $\DeclareMathOperator{\cesaro}{C}$ After reading Terry Tao's post on generalizations of the ...
6
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1answer
85 views

What does this number sieve have to do with pi?

Playing around with numbers I stumbled upon the sequence that begins $1,3,7,13,19,27...$ Looking it up on OEIS it is $A000960$ and is also known as the Flavius Josephus sequence. It is generated by ...
6
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112 views

$a_k = \sum\limits_{i=1}^{k-1}a_i \left(\frac{1}{k-i}-\frac{1}{k-i+1}\right)$ and $a_1 = 1$, prove the series is decreasing?

If the series {$a_n$} is defined recursively in the following way: $a_1 = 1$, $a_k = \sum\limits_{i=1}^{k-1}a_i\cdot\left(\frac{1}{k-i}-\frac{1}{k-i+1}\right)$ for $k=2,\cdots, n$, how can I prove ...
6
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107 views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^5 2^n}$

Given the nth harmonic number $ H_n = \sum_{j=1}^{n} \frac{1}{j}$, we get from this post that apparently, $$\sum_{n=1}^{\infty}\frac{H_n}{n^k}z^n= S_{k-1,2}(z) + \rm{Li}_{\,k+1}(z)$$ for $-1\leq z\...
6
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192 views

Compute $ \xi_p= \prod_{n=1}^{\infty} (1+\frac {1}{n^p})$

The main question I want to ask is inspired from this question Find the value of $$\prod_{n=1}^{\infty} \left(1+\frac {1}{n^2}\right)$$ Now, I have solved this question easily using product ...
6
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0answers
137 views

On $\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^4$ and Gieseking's constant

I. Intro While trying to solve this post about the function, $$F(k)=\sum_{n=0}^{\infty}{2n+3\choose n+1} \left(\frac{1}{2^n}\cdot\frac{3}{2n+1}\right)^k$$ for $k=3$, I found out Mathematica can ...
6
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1answer
266 views

Infinite summation formula for modified Bessel functions of first kind

I was trying to find a closed form for the integral $$4\int_0^{\pi/2} t \, I_0(2\kappa\cos{t}) dt \; ,$$ where $$I_{\alpha}(z) := i^{-\alpha}J_{\alpha}(iz) = \sum_{m=0}^{\infty}\frac{\left(\frac{z}{...
6
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2answers
73 views

Does $\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$ converge or diverge?

I'm having trouble figuring out if the following series converges or diverges. $$\sum_{n=1}^{\infty} \frac{3+(-1)^n}{n}$$ Here's my thinking: $$\frac{2}{n} \leq \frac{3+(-1)^n}{n}$$ Since $\sum_{n=...
6
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0answers
72 views

Construction involving regular polygons inside a circle

Let's make a construction involving regular polygons: ► First, we begin with a equilateral triangle, with side $\ell_3 = 1;$ ► After, we draw a square on the middle point each side of the initial ...
6
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0answers
104 views

Is $\cot(\cot(\cot(\cdots\cot1)\cdots))$ always defined?

Consider sequence $a_1=1$, $a_{n+1}=\cot a_n$. Is $a_n$ always defined? Numerical evaluation suggests this conjecture is true. I have proved a weak version of this question: for a fixed $n$ and $...
6
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57 views

Assume $x_n>0$,$x_n+\dfrac{4}{x_{n+1}^2}<3.$ Prove $\lim\limits_{n \to \infty}x_n$ exists and evaluate it.

My Solution Notice that $$x_n+\dfrac{4}{x_{n+1}^2}<3=3\sqrt[3]{\frac{x_n}{2}\cdot\frac{x_n}{2}\cdot\frac{4}{x_n^2}}\leq \frac{x_n}{2}+\frac{x_n}{2}+\frac{4}{x_n^2}=x_n+\frac{4}{x_n^2}.$$ This ...
6
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1answer
137 views

$\frac37x + \frac{3\times6}{7\times10}x^2 + \frac{3\times6\times9}{7\times10\times13}x^3+ \cdots$

Convergence of the series for $x>0$: $$\frac37x + \frac{3\times6}{7\times10}x^2 + \frac{3\times6\times9}{7\times10\times13}x^3+ \cdots$$ The general tem is $a_n = \frac{3\times6\times \cdots \...
6
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1answer
135 views

Generalized Infinite Integration by Parts

During my studies in calc 2, I became fascinated by the integral $\int e^{-x^2}dx$ after hearing from the professor that it has no elementary function as its integral. I came up with an interesting ...
6
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115 views

Proof on positive sequence with $\limsup_{n} a_n^{1/n}=1$ and $\liminf_{n}a_n^{1/n} <1$

Does a positive sequence $\{a_n\}$ with $\limsup_{n} a_n^{1/n}=1$ and $\liminf_{n}a_n^{1/n} <1$ must have a subsequence $\{a_{n_i}\}$ satisfying $\lim_{i} a_{n_i}^{1/n_i}=1$ and $\lim_{i} |a_{n_i}^...
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0answers
245 views

Finding a general formula for the sequence {$0, 1, 3, 5, 6, 7, 9, 15$}.

I'm working on a mathematical model which requires me to generalize the elements of an infinite set. The element n is the nth finite sequence of the set. For: ...
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0answers
131 views

Weird sum that is almost definitely not $\sqrt 2$

I have not the ability to compute more than four digits of $$\sum_{n=1}^\infty \frac{1}{n^2 H_n^{(\ln n)}}$$ $H_n^{(m)} = \sum_{k=1}^n \frac{1}{k^m}$ is the generalized harmonic number. I know ...
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0answers
316 views

Series whose Cauchy product is absolutely convergent - A general example

Is there series that is divergent or conditionally convergent with absolutely convergent Cauchy product? Seems like there is a group of these examples! Perhaps finding divergent series with ...
6
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86 views

Does there exist a series with property $n\sum_{k=0}^{\infty} u_{nk}=1$?

Shortly, the idea is to find such series which admits "lazy" calculation: instead of computing all the terms, it would be enough to calculate its even terms (the case with $n=2$), and then multiply ...
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96 views

Disproving the existence of a specific infinite sequence of Fibonacci primes

Consider the following sequence: $$ T_{1} = a,\: T_{i+1} = F_{T_{i}} $$ where $ a \in \mathbb{P} $ and $F_i$ is the $i$-th Fibonacci number. Is there a value of $a \neq 5$ such that this sequence ...
6
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205 views

Pointwise limit of continuous functions, but not Riemann integrable.

I am trying to find a simple example of a function $f:[0,1]\rightarrow\mathbb{R}$ which is a pointwise limit of continuous functions, but is not Riemann integrable. I know the classical example where ...
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0answers
95 views

Possible formula for $ f(x) = \sum_{n=0}^{\infty}x^{-n!} $

I was wondering if we have a formula for the following function: $$ f(x) = \frac{1}{x^{0!}} + \frac{1}{x^{1!}} + \frac{1}{x^{2!}} + \frac{1}{x^{3!}} + ... = \sum_{n=0}^{\infty}x^{-n!} $$ (Like we ...
6
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1answer
121 views

Limit at Infinity of Maclaurin Series

Let $f(x) = \sum_{n=1}^\infty a_n x^n$. What is $\lim_{x\rightarrow \infty} f(x)$ in terms of the $a_i$? That question may be too broad, so here are some restrictions: Assume f(x) is continuous (and ...
6
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0answers
291 views

Find an explicit formula for the recursive formula

Find an explicit formula for the recursive formula: $$a_{n+1} = 2a_n\left(a_n + 3\right); a_0 = 4$$ The first few terms in the sequence go like this: $4, 56, 6608, \dots$ After $a_2$ the sequence ...
6
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1answer
160 views

Consider the metric space of infinite sequences of 0s and 1s under this metric.

For $x, y ∈ \{0, 1\}^\mathbb{N}$, define $d(x, y) = 2^{-n}$ where $n$ is the first position where the sequences $x$ and $y$ are different. Show that ($\{0,1\}^\mathbb{N}, d$) is compact. Show that ...
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52 views

Finding invariant measures (to solve recurrences)

INTRO: If you don't feel like reading the justification for my question, just skip down to the question. Some recurrence relations can be solved by finding invariants, or quantities that remain ...
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0answers
86 views

The fractal dimension of the Kolakoski sequence is $2-1/e$

The Kolakoski sequence, which is defined as the infinite sequence of symbols {1,2} that is its own run-length encoding (Wikipedia), has been suggested to be self-similar$^{1}$. The fractral ...
6
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0answers
180 views

Alernative way of solving summation $5+55+555+\ldots$

Question Find the summation for the series-: $$5+55+555+5555+...$$ I know it is a duplicate of this, but still, I am posting this because i was thinking of solving another way due to which I got ...
6
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2answers
257 views

Closed form for $\sum_{n=1}^\infty \left(e-\left(1+\frac{1}{n}\right)^n \right)^2$?

Is there a closed form for this series: $$\sum_{n=1}^\infty \left(e-\left(1+\frac{1}{n}\right)^n \right)^2 \approx 1.273278374727530507449$$ (Mathematica computation by Patrick Stevens). This ...
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0answers
106 views

Limit points of a sequence constructed from pi (if pi is normal)

I'm interested in this question apparently posed by John Nash, which I found in the book A Beautiful Mind. If you make up a bunch of fractions of pi $3.141592\ldots$. If you start from the ...
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0answers
240 views

Finding a quadruplet of “cow-equivalent” integers

This a follow up question/puzzle from an earlier question here How does rounding affect Fibonacci-ish sequences? Explanation For this new question, one needs only to understand a particular ...
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135 views

A generalization of the Euler-Mascheroni constant

Let $f:[1,+\infty)\rightarrow \mathbb{R}$ be a differentiable function. We are dealing with the limit of the sequence $$ f(n)-\sum_{k=1}^nf'(k). $$ If $f=\log$, then it is convergent to $-\gamma$ (...
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239 views

$K$-functional between $\ell_1$ and $\ell_2$ for a specific sequence

Short version: For any $n\in\mathbb{N}$, Let $$ p_n(k) \stackrel{\rm def}{=} \frac{1}{(k+1)\ln(k+1)}, \qquad 1\leq k\leq n-1 $$ and consider $$ \kappa_{p_n}(t) = \inf\{ \lVert u\rVert_1+t\lVert v\...
6
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67 views

Explain approximate lines in graph of this function

Sorry that this is a long question; the crux of it is that I want to know why lines appear in the graph of the function ($\varphi^\infty(x)$) I've defined. Define $\varphi(x)$ as follows: If the ...
6
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0answers
138 views

Explicit definition of a sequence

Suppose there are 6 sequences $a=(a_n)_{n\geq 0}, b=(b_n)_{n\geq 0},c=(c_n)_{n\geq 0},d=(d_n)_{n\geq 0},e=(e_n)_{n\geq 0},f=(f_n)_{n\geq 0}$, the data can be seen here: Data. I found out by trial and ...
6
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0answers
84 views

Can conditionally convergent series be interpreted as a “generalized Henstock-Kurzweil integral”?

One amazing thing about the Lebesgue integral is that is defined w.r.t. to a given measure and that there a lot of different measures making the Lebesgue integration a very general tool (consider ...
6
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1answer
171 views

Chance of flipping 50 heads over a span of 100 flips given more than 100 flips

So I found that the chance of flipping 50 heads out of a string of 100 flips is $$0.5^{50} (1-0.5)^{50} \binom{100}{50},$$ My question is, how do the chances of having at least 1 string of 100 ...
6
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0answers
96 views

How to compute $\prod_{n=2}^{\infty} \left(1-\frac{1}{n^n}\right)$?

Does $$\prod_{n=2}^{\infty} \left(1-\frac{1}{n^n}\right)$$ have any closed form in terms of known mathematical constants?
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103 views

More on $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$

Let, $$\alpha=2\sqrt[3]{1+\sqrt2}-\frac2{\sqrt[3]{1+\sqrt2}}$$ In this post, it was asked if, $$\sum_{n=1}^\infty\frac{(4\,n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^...
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137 views

Accurate $\zeta(s)$ integral identities for $\sum_\limits{n=2}^{\infty}\frac{1}{n^{s}\sqrt{\ln{n}}}$

Some time ago while doing formal symbolic manipulations for fun (without worrying about convergence or getting into analysis) to see where I would get, I did the following manipulation: Starting with ...
6
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1answer
149 views

Is it possible to find aproximation of conformal map from sequences of complex points?

I want to find equation of conformal map (= Fatou function $\Psi : z \to u$ ) which: maps some region of complex plane ( attracting petal) to right half of complex plane in u coordinate $Re(u) > ...
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269 views

Division of a square and value of a disk

I cam across this problem and I really don't know how to solve it. So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the ...
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0answers
76 views

Show there are only a finite number of integers with $\dfrac{\prod_{i=1}^n a_i-1}{\prod_{i=1}^n (a_i-1)} $ an integer

Show, for each $n$, there are only a finite number of integral $(a_i)_{i=1}^n$ such that $2\le a_i \le a_{i+1}$ and $\dfrac{\prod_{i=1}^n a_i-1}{\prod_{i=1}^n (a_i-1)} $ is an integer. My question is ...
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651 views

Show that $f(z)=\sum_{n=0}^{\infty}z^{2^n}$ can't be analytically continued past the unit disk.

I'm reading the problems of Stein and Shakarchi's Complex Analysis, Chapter 2 Problem 1 asks to show that $$f(z)=\sum_{n=0}^{\infty}z^{2^n}$$ cannot be analytically continued past the unit disk. (...
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67 views

Prove $\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kx)\,dx$

Let $N>0$ and $a_0,a_1,...,a_N$ be any positive numbers. How to prove that $$\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(...
6
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79 views

Estimation of a sequence related to the Stirling's formula

I need to show that $$n!=\left(\frac{n}{e}\right)^n\sqrt{2\pi n}e^{\lambda_n}$$ where $$\frac{1}{12(n+1)}<\lambda_n$$ I calculated that $$\lambda_n=\ln n!+n-n\ln n -\frac{1}{2}\ln(2\pi n)$$ On ...
6
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142 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
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166 views

Limit superior, limit inferior and a series involging $\sum_{k\nmid n}$k, where $1\leq k\leq n$

The purpose of this post is state assertions by the use of statements and hypothesis in an expository way and after I am asking for reasonable unconditionally results that you can provide us. Using ...
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150 views

Sequence involving floor function - limit and bounds

My first time here, so please excuse any breaches of etiquette. For a given $p \in \mathbb N$ and irrational $\alpha$, let $\varepsilon_n=\alpha n-\lfloor \alpha n \rfloor,$ and $S_n=\frac{1}{n+1}\...