For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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144 views

A connection between a sequence from the Collatz conjecture and a sequence of densities from $\zeta(k)-1$?

Just for grins, I created lists of first-entries of finite sequences of rank $r$ for the Syracuse problem (Collatz conjecture using only odd numbers) and found these sequences on OEIS. My sequences, ...
2
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52 views

Distribution of Omega values modulo m

Define $\Omega(2^{a_1}3^{a_2}...p_k^{a_k})=a_1+...+a_k$ . I am interested in the density of the values of Omega mod m. If we define the set $S=(x:\Omega(x)\equiv k \text{ mod m})$, I would like to ...
2
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53 views

Bound on the difference of two convergent infinite products

Let $(\alpha_n)$ and $(\beta_n)$ be two sequences of non-zero complex numbers such that the products $\prod_n \alpha_n$ and $\prod_n \beta_n$ are convergent. How to prove the following inequality? $$ ...
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77 views

Simplify series involving derivatives

In order to get the cosine transform of a Marcum Q function of order 1 (see this), I ended up with this series: ...
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85 views

A property of a sequence of summable sequences

Let $(a_n)$ be a summable sequence of positive real numbers then we can find a sequence $w_n\to \infty$ such that the sequence $(a_nw_n)$ is still summable. This property has been asked and answered ...
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157 views

Finding the sequence of $n$ length where it is possible to make every value from one to as high of an integer as possible by adding adjacent values

Is there any easy way of finding the sequence of $n$ length where it is possible to make every value from one to as high of an integer as possible by adding adjacent values (lone numbers are allowed ...
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46 views

Infinite series with only two zeros at $\Re(s)=\frac12$. Why is that the case?

I was experimenting with the following series: $$\displaystyle k(s)=\sum_{n=1}^\infty \frac{1}{n^{\ln (\frac{n}{s})}}$$ I believe this is an entire function without any poles in $\mathbb{C}$ (note ...
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77 views

Absolute convergence series laws II (help with a proof)

Hi everyone this is easy I think. My question is regarding to the next proof. [I've shown that if $\sum _{x\in X}f(x)$ and $\sum _{x\in X}g(x)$ are both absolutely convergent then $\sum _{x\in ...
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94 views

Troublesome proof in Functional Analysis with dual vector space

Greetings to all of you I have tried to prove the following theorem but I am having some troubles with it. Let $X$ be a separable normed space and $(x_n')$ a bounded sequence in $X'$, then there is a ...
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130 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
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29 views

what is the third term of the $\omega^{\omega}$-accelerated squares?

let $\{s_n\}_{n=0,1,2...}$ be a strictly increasing sequence of positive integers, indexed by the finite ordinals. if $S$ denotes the space of all such sequences, let the accelerator function, $\alpha ...
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111 views

necessary and sufficient conditions for the following inequality to be hold

Let $p=(p_n)$ is a sequence of nonnegative numbers with $p_0>0$ s.t $P_n= \sum_{k=0}^{n}p_k \rightarrow \infty$ as $n \rightarrow \infty$ and let $t\in(0,1)$. find necessary and sufficient ...
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87 views

Behavior of $\sum_{k=a}^{b} {\frac{1}{k(\log k)(\log \log k)\cdots(\log^n k)}}$ under limits.

Define $$S_n (a, b)=\sum_{k=a}^{b} {\frac{1}{k(\log k)(\log \log k)\cdots(\log^n k)}}$$ where $\log^n$ denotes $n$ compositions of the natural logarithm. And $$P(n>2)=\lfloor ...
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2k views

Compute the first five non-zero terms of the Taylor series about $a=4$ for $f(x)=\sqrt{x}$

This is my first Taylor Series problem and I want to make sure I completed it correctly. Here is the question: Compute the first five non-zero terms of the Taylor series about $a=4$ for ...
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35 views

Radius of Convergence and Interval of Convergence: Am I doing it right?

I must find ROC and IOC in $$ \sum_{n=1}^\infty \frac{{(-1)}^n(x^{2n})}{\sqrt[3]{n^2+4n}} $$ I get $$ R= \lim_{n\rightarrow \infty}\left| \frac{a_n}{a_{n+1}} \right| = \cdots = \lim_{n\rightarrow ...
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118 views

Series convergence Question ( TIFR GS $2010$)

Question is : What i have done so far is : I see that $\frac{\pi}{n}\rightarrow 0$ and so should be the sequence $u_n=\sin (\frac{\pi}{n})$. i.e., $u_n$ converges to $0$ and so $(b)$ is true. I ...
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107 views

Need to find a better algorithm to solve a project euler problem dealing with coprime pairs.

I've been working on this for a while and found several solutions so far, but none are fast enough to find the necessary $S(10^7)$. Here is the question: For an integer $M$, we define $R(M)$ as ...
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50 views

Calculate the Maclaurin series, using binomial series

Calculate the Maclaurin series for the following function This is a note from my teacher through email "Question 5a (this question) is now a bonus question, since it requires binomial series that we ...
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73 views

What is $iav-\log(v)$? Any series expansion or inequality for it?

I am investigating the integral of this question here where \begin{equation} \frac{\exp(i a v)}v=\frac{\exp(i a v)}{\exp(\log(v))}=\exp(iav-log(v)) \end{equation} where I am interested in the ...
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110 views

Is there a “natural” norm in the space of all sequences?

Let $X$ be $\mathbb{R}^\mathbb{N}$, the space of all real-valued sequences. Is there a natural norm in this space?
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55 views

Structural differences between closed forms of two related infinite products?

I have asked this question on MO, but so far did not attract a single response. Take $a \in \mathbb{R}, s \in \mathbb{C}$ and this infinite product of conjugated zeros: $$\displaystyle C(s,a) := ...
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89 views

Finding every $\{a_n\}$ such that $\sqrt{a_{2n-1}+a_{2n}}\in\mathbb N, a_n=d_{n-1}d_n$ where $d_n=(a_n, a_{n+1})$

Question : Find every possible pair $(k,l,m)\in\mathbb Z$ such that the following $(1),(2)$ hold where $$a_0=a_1=1, a_{n+2}=ka_{n+1}+la_n+m\ (n\ge 0).$$ $(1)$ $a_{2n-1}+a_{2n}\gt 0$ is a ...
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178 views

Probability of having a path of a given length in a random graph

Suppose $G=\langle V,E \rangle$ is a directed graph consisting of $n\in \mathbb{N}$ vertices. Vertex $v_i \in V$ has an edge to vertex $v_j \in V$ with a probability of $P(i, j) = f(|i-j|)$ where $f$ ...
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811 views

Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{ 2^2}+\cdots+\frac{1}{n^2}$

Use Cauchy Criterion to prove the convergence of $x_n=1+\frac{1}{2^2}+\frac{1}{3^2}+ \ldots +\frac{1}{n^2}$ My attempt Take $|x_m-x_n|$, where $m>n$, We have ...
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116 views

What's convergence in a group look like?

How should we define convergence for sequences and series in groups? Here's maybe how to do it: Let $G$ be a group. A norm will be like a norm defined on a vector space except we'll define it with ...
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29 views

Decomposability of a function into two

This is question on the existence of a decomposability of a function $f(v)$ into multiplicative factors $\lambda(v)$ and $g(v)$. The question is non-trivial since the functions $\lambda(v)$ and $g(v)$ ...
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685 views

Getting K heads out of N biased coins problem (formula generation ).

Problem- Given a set of coins n with each coin i having Pi probability to give heads. Find the probability of getting k heads, when all coins are tossed together. hi i have solved this problem ...
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113 views

Convergence of partial sums of an alternating series

Suppose $x_0, x_1, x_2,\ldots$ is a sequence of positive numbers monotonically converging to zero. Then $x_o - x_1 + x_2 -x_3+\cdots$ converges. How would you prove this statement? I know you would ...
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51 views

For decreasing $a_n > 0$, for what increasing $f:{\mathbb N} \to {\mathbb N}$ does $\sum_n a_n$ converge iff $\sum_k f(k) a_{f(k)}$ converges?

A classical result says that for a positive decreasing sequence $a_n$, the series $\sum_n a_n$ converges if and only if $\sum_k 2^k a_{2^k}$ converges. From the proof, it seems that we could replace ...
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45 views

Looking for the General solution of a series

What is the general solution (for $f(t)$) of the following series? $$f(t=-1) = \text{Does-not-apply !}$$ $$f(t=0) = 1$$ $$f(t=1) = 1 + \frac{x-y}{2}$$ $$f(t=2) = 1 + \frac{x-y}{2} + ...
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114 views

Case study in population biology. Sequences and series, find general solution.

Think about a population of individuals which all have two set of chromosomes (as human do). There is one gene that codes for a helping behaviour. This gene has two alleles (an allele is a variant of ...
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167 views

Does this series that has terms $1/n$, then terms $1/n (\log n)^2$, then terms $1/n \log n (\log \log n)^3$, etc. converge or diverge?

Define $\log_{(k)}$ to be the logarithm function iterated $k$ times, where $\log_{(0)}$ is the identity function. Consider the series $\sum_n 1/a_n$ where $$a_n = (\log_{(f(n))} n)^{f(n)+1} ...
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185 views

Is it possible to find sum of this series?

I am trying to find the sum of the following series asked by my friend. $$n\cdot\left(\bigl\lfloor\tfrac{n}{2}\bigr\rfloor+ \bigl\lfloor\tfrac{n}{3}\bigr\rfloor+ \bigl\lfloor\tfrac{n}{4}\bigr\rfloor+ ...
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77 views

Taylor expansion proof

It's pretty clear to me that in this expansion: $$p(x) = f(0)+f'(0)x+f''(0)\frac{x^2}{2!}+f'''(0)\frac{x^3}{3!}+\cdots$$ When I assume $x=0$, $p(0)$ is gonna be equals to $f(0)$ and all its ...
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81 views

Upper Bound on $\frac{1}{1-\beta u}-\sum\limits_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$

Is there any procedure to find an upper bound of the following expression? $$\frac{1}{1-\beta u}-\sum_{n=0}^{\infty}\frac{e^{-u}u^n}{n!(1-\beta(n+1))}$$ Here $u,\beta\in\mathbb{R},\ u>1,\ ...
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59 views

Bounds on a rapidly increasing sequence

I read about a sequence similar to this one here on Stack Exchange a while back, somebody used it as an example for something that I can't recall! However, when I read about it it made me come up with ...
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47 views

question on uniform convergence

Assume $p_0=0$ and for all $n\in\{0,1,2,....\}$ we define $P_{n+1}(x)=p_n(x)+\dfrac{x^m-p^m_n(x)}{2}$ then how should I find all natural numbers $m$ such that $\{p_n(x)\}$ be uniformly convergent ? ...
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125 views

Prove that $\arctan\Big(\frac{100 \log^2 n}{\sqrt{n+1} - \sqrt[3]{n}}\Big)$ is decreasing

I would like to prove that $\exists n_0$ such that the sequence $a_n = \arctan\Big(\frac{100 \log^2 n}{\sqrt{n+1} - \sqrt[3]{n}}\Big)$ is decreasing $\forall n \ge n_0$. It is sufficient to show that ...
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197 views

Showing the relationship between Lucas and Fibonacci numbers with a recursive relationship

Given the equation $$L_{n+k}-(-1)^kL_{n-k}=5F_kF_n$$ Where $\{L\}$ is the set of Lucas numbers and $\{F\}$ is a Fibonacci sequence, $n$ and $k$ are both $>0$ and integers, how would one develop ...
2
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92 views

Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene

Let $x=(x_n)$ be a Cauchy sequence in a metric space $X$ having a convergent subsequene $x\sigma=(x_{\sigma_n})$ where $\sigma:\mathbb N\to\mathbb N$ is strictly increasing. Then $(x_n)$ is ...
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86 views

Name for this Sum

This is probably a very basic question, but I am asking because I could not find the answer online. I have been trying to find out some properties regarding the following sum ...
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51 views

Sequence of Ratios

Let $\{a_n\}_{n\ge 0}$ be a positive real sequence and define $$r_n=\frac{a_{n+1}}{a_n},\quad n\ge 0$$ Suppose that we know the formal power series of $a_n$, i.e. we know the following: ...
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79 views

Supposing $a_{n+1}={a_n}^2+m \ \ (n=1, 2,\cdots)$, represent $a_n$ by $a_1$ and $n$.

Supposing that $m$ is an integer and that$$a_{n+1}={a_n}^2+m \ \ (n=1, 2,\cdots),$$ represent $a_n$ by $a_1$ and $n$. I'm interested in this question because I got the following. If $m=-2$, then ...
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192 views

How to fulfill those boundary conditions?

The problem is the following: Think of a set of functions depending on spherical coordinates given by: $${\psi}_{l m}(r,\theta,\phi) ={k_l}(ar) P_{l}^{m}(\cos \theta) e^{\pm i m\phi} ,$$ so ...
2
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78 views

Simplify this double series

if I have a double sum and I have an expression like $$ \sum_{l=0}^{\infty} \sum_{l'=0}^{\infty} g(l)f(l') \frac{1+\cos(\pi(l+l'))}{1+l+l'},$$where g and f are some functions. The thing is: I could ...
2
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44 views

Evaluation of a multiple sum involving $\min\{i_0, i_1+ \cdots+ i_n\}$ with $i_1+ \cdots+i_n\leq x$

How can I calculate $\displaystyle\sum_{i_0=0}^x \sum_{i_1,\ldots, i_n=0}^1I_{i_1+ \cdots+i_n\leq x}\min\{i_0, i_1+ \cdots+ i_n\}$ as a function of $n,x$? $I_{i_1+ \cdots+i_n\leq x}$ is ...
2
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0answers
211 views

Prime number finding via polynomials

I try to find approximation polynomial to estimate which number is prime or not. Addtion to this, (If It is possible) To find the closed form of coefficients of the series ($c(n)$) Euler found the ...
2
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182 views

Entropy Rate of a sequence of Random Variables with Distributions related to Primes

Let us consider a stochastic process $\mathcal{X}=\{X_i\}_{i \in \mathbb{N} }$ where $X_i$'s are independent and $X_i$ is distributed as $$X_i=p_k \ \mbox{w. p.}\frac{p_k}{\sum_{l=1}^{i}p_l},\ 1\leq ...
2
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0answers
88 views

Converting from Closed Form

Let $A(n) = \lfloor n/2+\log_2(n)-\log_2(2) \rfloor$. Is there an easy way to convert this closed form into a recursive form? If so, what is the general method, and how might it be applied here. If ...
2
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0answers
85 views

Finding the general term of $x_n$

Find the general term of $(x_n)_{n\in\mathbb ,N}$, $a>0$, defined by the recurrence relation $$x_{n+1}=\dfrac{1}{2}\left(x_n+\dfrac{a}{x_n^2}\right), $$ EDIT - consider the case $x_0>0$