For questions about recurrence relations, convergence tests, and identifying sequences

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Recursive sequence with a formula for a part of its criteria

I have the next recursive sequence which firts terms are $$2,\ \frac{3}{2},\ \frac{10}{7},\ \frac{17}{12},\ \frac{112}{89}$$ I need to express it has a general form for the $n$th element, I can't make ...
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34 views

Is a sequence always defined by a formula holding for all values?

Suppose we have the sum of a series $S_n = T_1 + T_2 + \cdots + T_n, S_1 = 6, S_2 = 20, S_n = 6S_{n-1} - 8S_{n-2}$. The explicit formula for the sum can then be derived as $S_n = 4^n + 2^n$. Yet when ...
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20 views

Prove the uniform convergence of series in a compact space

I have to prove that if $\sum\limits_{n=1}^\infty \frac{1}{|a_n|}$ converges, then $\sum\limits_{n=1}^\infty \frac{1}{x-a_n}$ converges absolutely and uniformly in any compact space that $\forall n ...
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37 views

Question about series and how the pattern idea works

Two Questions: When you are given: $1, 2, 3, .... , n$ How do you know that in the $...$ that it continues the $x_{n-1} + 1$ pattern? Is it the definition of series? Secondly: Do partial sums ...
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74 views

Simple finite series with reciprocal factorials

I'm trying to find the following sum: $$ \sum_{k=0}^n{1\over(n-k)!}{x^{k+2}\over k+2}. $$ The most obvious way is to differentiate wrt to $x$ leading to $$ ...
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27 views

Does such formula exist $\lim_{n\to \infty}\frac{\log_ex_n}{x_n-1}=1$

Does such a formula of limit related to sequences exist? $$\lim_{n\to \infty}\frac{\log_ex_n}{x_n-1}=1$$ where $x_n$ is a sequence .
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28 views

Does $\lim \frac {a_n} {b_n}$ exist and $\lim a_n \neq 0$ imply $\lim b_n$ exist?

Suppose $\lim_{n \rightarrow \infty} \frac {a_n} {b_n}$ exist and $(a_n)$ converges to some number $k \neq 0$. Is it then possible to conclude that $(b_n)$ converges ? Also, suppose $\lim_{n ...
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58 views

Convergence of series, cesaro summability

Suppose that series $\sum_{n=1}^\infty na_n^2<\infty$ and $\sum_{n=1}^\infty a_n$ is Cesaro summable. How do you show that $\sum_{n=1}^\infty a_n$ converges? I know that if $\lim_{n\rightarrow ...
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32 views

Does the series $\sum_{n=1}^\infty \frac{(-1)^n}{n^{1+\frac{1}{n}}}$ converge?

Attempt: We can write the terms in the series as $(-1)^n a_n$ where $$ a_n = \frac{1}{n^{1+\frac{1}{n}}}< \frac{1}{n}.$$ And since $\lim_{n \to \infty} \frac{1}{n} = 0$ and is monotonically ...
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47 views

Separable dual space implies existence of complete series

Let $\{x_n\}$ be a basis for a Banach space $X$ and let $\{f_n\}$ be the associated sequence of coefficient functionals. Prove or disprove: if $X^\ast$ is separable, then ${f_n}$ is complete ...
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63 views

How to calculate the value of the series

Evaluate the following series $$\sum\limits_{n = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 1}}}}{n}\left( {\sum\limits_{k = 1}^n {\frac{1}{k}} } \right){{\left( {\sum\limits_{k = 1}^n ...
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44 views

How to calculate the value of the series limits

$$\mathop {\lim }\limits_{x \to {1^{\rm{ - }}}} \left\{ {\sum\limits_{n = 1}^\infty {\frac{{\widetilde{H_n^3}}}{{n + 1}}} {x^{n + 1}} + {{\ln }^3}2\ln \left( {1 - x} \right)} \right\} = ?.$$ where ...
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44 views

Problem showing a double summation equality.

I'm trying to show that $$G(L) = \sigma^2(\sum_{j=0}^\infty \psi_{j}^2 + \sum_{h=1}^\infty\sum_{j=0}^\infty \psi_j \psi_{j+h}(L^h-L^{-h}))$$ is equal to: ...
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51 views

Trying to find sum of a complex infinite series with Gamma function and factorials

I am trying to find the sum $S$ of the following series. $$S = \sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}t^{2n}\Gamma\left(\frac{1 + 2nH - ...
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30 views

Is my convergence proof correct?

It's "obvious" that the following sequence converges. I was asked to prove it on a homework assignment and was given no credit for my proof. I wanted to ask the community here if 1) they think my ...
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55 views

Double series of Harmonic Numbers

In a solution presented here a series involving the product of Harmonic numbers is involved. The intent of the problem is to determine a form of the series \begin{align} \sum_{n=1}^{\infty} ...
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93 views

Inequality involving a strictly positive sequence

I was asked to prove the following : \begin{array}{l} x_n > 0,\,\forall n \in Z^ + \\ \lim \sup (\frac{{x_{n + 1} + x_1 }}{{x_n }})^n \ge e \\ \end{array} Is my approach correct ? Using ...
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93 views

Has anyone seen this form of the Collatz Conjecture?

This question asks if this form of the Collatz Conjecture has been reported or is all ready known. The goal of this question is to determine if I should write a paper on it's discovery or not and ...
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52 views

Can one prove the divergence of $\sum \frac{1}{p}$ by the absolute convergence criterion of infinite products?

Euler proved this celebrated theorem that $\sum \frac{1}{p}=\infty$ by using the product formula that $\displaystyle \zeta (s)=\prod \left( \frac{1}{1-p^{-s}}\right)$. Now I thought of another ...
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34 views

Which $L \subset [0,1]$ equal the set of limits of a sequence of a sequence in $[0,1] \setminus L$?

I was glanced at this question here and it cause me to wonder the following: Question: Is there a simple description of the subsets $L \subset [0,1]$ with the property that there exists a sequence ...
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26 views

Approximating the max value of a function containing the half-sum of binomial series

I recently met a problem and I have been finding related materials about it. However it is still hard to handle. Precisely, I want to maximize a function related to $$S_n = \sum_{i=0}^{\lfloor ...
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45 views

tricky question regarding Series, Limits and Convergence

Suppose we have 2 real power series for every $k=1,2,3,\ldots$ $$ f_k(x)=\sum_{n\in\mathcal{S}^{(+)}(k)} x^n d_n(k),\quad g_k(x)=\sum_{n\in\mathcal{S}^{(-)}(k)} x^n d_n(k), \quad d_n(k)\geq 0 $$ where ...
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Convergence of a Double Sum over 2 integers

Does the following double summation over x, x' (both integers) converge? $\sum\limits_{x=-\infty}^\infty \sum\limits_{x'=-\infty}^\infty \frac{Sin^2(2 \pi(x-x'))}{(x-x')^2}$. If so evaluate the sum. ...
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43 views

Sort of Convolution

I was wondering if the following convolution I am considering already has a name or well-studied. Thanks for your help. If the condition of the positive radius of convergence is not enough, then I ...
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86 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
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24 views

Prove that this sequence of fractions is returned unchanged after the divisor recurrence, the matrix inverse, and the sum over divisors.

Consider the fraction of binomial coefficients and powers of $4$: $$a(n)=\frac{\binom{2 (n-1)}{n-1}}{4^{n-1}}$$ starting: ...
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21 views

Sequence terms being divisible

Here's a question I would like hints for: The sequence ${x_n}$ is defined by $x_{n+2}=6x_{n+1}-9x_{n}$ for $n \geq 0$ where $x_0=3$ and $x_1=18$. What is the smallest $k$ such that $x_k$ is ...
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If $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$

I tried proving that if $\lim_{n \to ∞} z_n=L_1$ then $\lim_{n \to ∞} \overline{z_n}=\overline{L_1}$ for $z_n \in \mathbb{C}$. This is my attemt. Let $\epsilon>0$. Then there exists $N\in ...
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34 views

Problems with convergence in mean

I hope you can help me with the following problem Let $\{ e_i : i\in \mathbb{Z}\}$ be and independent U.I. sequence of scalar random variables with zero mean. Let $\{ A_j : j \geq 0\}$ be a sequence ...
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price pool distribution with a series

I'm trying to make a small program, that gives me price distributions for tournaments and I thought using a series would be good for that. I found this formula: $$\sum_{k=1}^{n} ar^{k-1} = ...
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44 views

Expectation of $\frac{1}{X+1}$ for a geometric random variable

I am confused over $E(\frac{1}{1+X})$ where $X$ is geometric distribution with parameter $p$. The book wants me to prove that $E(\frac{1}{1+X})=log((1-p)^{\frac{p}{p-1}})$ Here's what I did. ...
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49 views

What is the most “powerfull” method to prove a sequence is increasing or decreasing?

Given a sequence $a_n$ defined in a recursive manner, the methods I know to prove if the sequence is increasing are: 1) observe if $a_{n+1} - a_n > 0 \ \forall n.$ 2) take $\frac{a_{n+1}}{a_n}$ ...
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92 views

Double summation of a geometric series

I am interested in the following sum for a given value of n: $ \sum\limits_{x=1}^{n} \sum\limits_{y=1}^{n}x^y$ I can simplify this to $ \sum\limits_{x=1}^{n} \frac{x^{n+1} - x}{x - 1}$ From here ...
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Boundedness of sequence

Consider a dynamical systems over complex numbers $$ z_{n+1}=\frac{\alpha}{z_{n}}+ \frac{\beta}{z_{n-1}},\qquad n=0,1,\ldots $$ where the parameters $\alpha, ~\beta$ are complex numbers, and the ...
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61 views

convergence of a series, exponent involving $\cos k$

In an answer to this question the problem was raised of the convergence or divergence of $$\sum_{k=1}^{\infty}\frac{1}{k^{2+\cos k}}\ .$$ The problem was (quite properly) dismissed as being ...
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57 views

Finding Function Representation of Recursive Sequence

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by ...
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59 views

Particles reproduce after a delay, what is the population after $m$ months?

I have been working on the following puzzle: A seed particle produces 'r' particles/month. Each of these particles must wait at least 's' months before producing 'r' particles/month. ...
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52 views

convergence of a sequence of cauchy

I would like to ask something about the convergence of a Cauchy sequence in a space $X$ metric. There will be a metric space $X$ such that If $(x_n)$ cauchy sequence in $X$ then $(x_n)$ is not ...
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Is there a closed-form expression for this matrix power series?

I encountered a matrix power series: $$ X = M + PMP^{t} + P^{2}M(P^{t})^{2} + \cdots, $$ where $M$ is a real symmetric matrix, and $P$ is a real square matrix. Assuming that this series converges, ...
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Find a closed form for the constant term

In a previous question, an asymptotic expansion was provided for the weighted divisor summatory function $\displaystyle \frac {d(n)}{n}$: $$\sum_{n\leq ...
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147 views

What is the value of the power series with rising factorial coefficients?

The power series given by \begin{equation} S(\alpha,n,x) = \sum_{k=0}^\infty a_k(\alpha,n)x^k,\quad n\in \pmb{N}_0,\quad x\in\pmb{R} \end{equation} where \begin{equation} a_k(\alpha,n)=\left[ \alpha ...
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Powerful numbers in Pell solutions (or, more generally, any Lucas sequence)

There are several definitive results regarding perfect powers in the Pell numbers — e.g., the only perfect power is $P_7=169=13^2$. On the other hand, when it comes to powerful numbers, I've only ...
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36 views

A question on infinite series and boundedness of sequence

Let $(a_n)$ be a real sequence such that for every convergent real series $\sum x_n$ of positive terms , $\sum |a_n|x_n$ is also convergent , then is it true that $(a_n)$ is a bounded sequence ?
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102 views

A function equal to its Taylor series on an interval is also equal to its Taylor series on a subinterval with different center

Suppse the power series $ \sum_{n=0}^\infty a_n (x-a )^n$ has positive radius of convergence $R$ and thus defines a real analytic fuction $f$ on $(a-R,a+R).$If $x_0$ is a point with $|x_0-a|<R,$ ...
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54 views

Series of polynomials and uniformly convergence

It's part of the proof of a Lemma of an article I was reading (Algebraic values of transcendental functions at algebraic points). I couldn't understanding one thing: Let f be a complex function such ...
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61 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
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70 views

sum of reciprocal power-1

I found this in my old notebook $$\sum_{n \text{ perfect power}} {\frac{1}{n-1}} = 1$$ and this was my "proof" $$ \begin{align} \frac{1}{1}+\frac{1}{2}+\cdots ...
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60 views

Understanding Sobol sequences

Can someone explain to me in simple terms, how Sobol sequences work? The wikipedia article is fairly technical. They look pretty interesting. So I shall describe (whatever little I know) in short the ...
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48 views

Sequence of primes by concatenating digits in a given base.

Given a base, $b$ is there is a sequence $\lbrace a_n\rbrace_{n\geq 0}$ where $a_k \in \lbrace 1,2\cdots, b-1\rbrace$so that the sequence: $$b_n:= \sum_{k=0}^n a_kb^k$$ is a sequence of primes ...
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53 views

Differentiation and integration of a series

If I have a power series $$\sum _{k}^{\infty }f(x)$$ and I differentiate it I get according to my current knowledge $\sum _{k}^{\infty }f(x)'$,however when I look at a power series defined by $$\sum ...