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

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Other Patterns in Triples

I have the following 20 triples generated by polynomial distribution: $$\begin{matrix} (2,4,5)&(2,3,4)&(2,3,5)&(1,4,5)&(2,2,4)\\ ...
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Closed form for binomial coefficient series

If $6|n$, is there a closed form for $$\sum_{t=\frac{n}{2}}^n\binom{\frac{n^2}{3}}{t}\binom{\frac{2n^2}{3}}{n-t}?$$
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Notation confusion about sum of $\Lambda (n)$

This is hopefully a small point of notation I am missing. I am used to the first two equalities below. $$\sum_{n \geq 1} \Lambda(n) n^{-s} = \sum_{p \mbox{ prime}} \sum_{m \geq 1} \Lambda(p^m) ...
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Solution Verification for a Convergence Question

The Question Give an example of a pair of series $\sum a_n$ and $\sum b_n$ with positive terms where the limit as n goes to infinity of $\frac{a_n}{b_n} = 0$ and $b_n$ diverges and $a_n$ converges. ...
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37 views

Application of Weistrass M- Test

Suppose that the series $\sum_{1}^{\infty}n|b_n|$ converges. Show that the series $\sum_{1}^{\infty}b_n \sin{nx}$ converges uniformly on R, and that it can be intergrated and differentiated term by ...
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is there anyone able to develop this series in order to get the following equality?

$\sum_{i=1}^\infty (1-\alpha)_{(i-1)}*\frac{\varepsilon^i}{i!}$ = $\frac{1-(1-\varepsilon)^{\alpha}}{\alpha}$ where $(1-\alpha)_{(i-1)}$ is the Pochammer symbol or rising\ascending factorial. Can ...
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44 views

Closed-form expressions of $\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$

Does anybody know if there's a closed-form expression of this series? $$\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$$ where $a$ and $b$ are strictly positive. It's easy to see that it's ...
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Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$?

Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$? If so, where can I find the equivalent of a Wikipedia entry?
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32 views

Series in closed form

Let $\alpha \in (0,1)$ and $z\geq 0$. Define the function: $$ \theta_{\alpha}(z)=\sum_{j=0}^{\infty}\left(\frac{z}{\alpha j + 1}\right)^j. $$ Can $\theta_{\alpha}(z)$ be written in closed form? Are ...
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When does the limit of the ratio of consecutive terms of a sequence exist?

I am trying to understand and obtain some sufficient conditions under which the limit of the ratio of consecutive terms of a sequence exists. Let $x_n$ be a sequence of positive integers, such that ...
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finding a function whose values at certain points are defined

I have certain polynomials in a sequence from which I am trying to derive the common term. The polynomials are: $7k^2-8\\ 21k^4-368k^2-704\\ ...
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Is there a such thing as a quasi-random shuffle?

I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce ...
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37 views

To find the approximate solution of the series for large N

to make sum of series including combinations ${N\choose 1}{N\choose 0}+{N \choose 2}{N\choose 1}a^2 b^{-2} + {N\choose 3}{N \choose 2}a^4 b^{-4}+{N\choose 4}{N \choose 3}a^6 b^{-6}+...$Is it possible ...
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Can Wynn's $\epsilon$ algorithm be used for sequence limit?

Let's assume we have a sequence $(a_n)$, which converges to some limit $L = \lim_{n\to\infty} a_n$. However, we are able to calculate only first $N$ terms of the sequence. It is clear that, in ...
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close form solutions for infinite sums

I am interested in finding the following infinite sums in order to reparameterize a distribution function. $\sum_{n=0}^{\infty} \theta^{n}q^{n(n-2)/2} $ where $\theta>0$ and $1>q>0$ and ...
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Solution Verification for Radius of Convergence of $\sum\frac{n^2x^n}{2\cdotp 4\cdotp 6 \cdots 2n}$

The Question: Find the radius of convergence and the interval of convergence of $\displaystyle\sum\frac{n^2x^n}{2\cdotp 4\cdotp 6 \cdots 2n}$ My Work $$\left|\frac{a_{n+1}}{a_n}\right| ...
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identifying expansion series

Consider $$ f(x;a)=\sum_{k=0}\frac{x^k}{k!}\prod_{i=0}^k[(1+a)^{i+2}-(i+2)a-1], $$ where $a>0$. Can the series in the right-hand side be reduced to (or is it) to a more compact form (e.g., a ...
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26 views

How many pure trees with a fixed number of nodes exist?

How many pure trees of size (number of nodes) $n$ exist? Apart from having this fixed size, the trees can be arbitrary. The sequence starts like this: Here's the beginning of the sequence:
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21 views

If I use a Hermite-Gauss function as a basis which functions can I represent?

I know that Hermite polynomials are orthogonal with eachother as follows: $$\langle H_n,H_m \rangle=\int_{-\infty}^\infty H_n(x) H_m(x) \exp(-x^2) \,\mathrm dx$$ If I define a basis function (the ...
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On a summation manipulation

I have $R_t= \frac{1}{h} \sum_{j=0}^{h-1} E_tr_{t+j} + \theta_t$ where $E_tr_{t+j} = E[r_{t+j}| I_t]$. By subtracting $r_t$ from both sides and after some manipulations I should get: $$R_t - r_t= ...
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Existence of a sequence with prescribed limit and satisfying a certain inequality II

This is a more specific variation of the question in the post Existence of a sequence with prescribed limit and satisfying a certain inequality Suppose you have two infinite sequences ...
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Multiplying two sums?

(Real-analysis only) I will admit, I have posted a question similar to this, but this question's aim is to ask how to multiply the sum and integrate it. $\displaystyle \log^2(x) = ...
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59 views

How to find $\sum_{k=1}^n k^k$?

Actually question which I found: Find the sum of the series $1^1+2^2+3^3+ \cdots +n^n $ This question has been bothering me since a long time. Any help would be appreciated!
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Solving an integral (or series) equations system

Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ...
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73 views

convergent and divergent series indeterminate by ratio and root test

(i) Give an example of a divergent series $\sum\limits_{n=1}^{\infty} a_n$ of positive numbers $a_n$ such that $$\lim\limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim\limits_{n \to \infty} a^{1/n} = ...
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$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$ is differentiable?

Let $ D=\{(x,y) \in \mathbb{R}^2 : x>0, y>0\}$. Show that the function $$f(x,y)={1 \over x^2} \sum_{n=1}^{\infty}{\int_x^y{\sqrt{t} \over {1+ ({t \over x} -n)^2}}} dt$$ is well defined on $D$. ...
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How to find the value of $ \sum_{n=0}^{1947} \left(\frac 1{2^n+ \sqrt{2^{1947}}} \right) $

I've no clue where to start from. I tried writing down a few terms to find out a pattern but didn't notice any. Any help would be appreciated! :)
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How much freedom do we have when working with sequences?

Consider the sequence $$a_n = \sum_{k = 1}^n \frac{1}{k} - \log(n)$$ and suppose we are to show it is Cauchy. (Do not solve the exercise for me. Spoilers are not welcome.) While there maybe are better ...
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Does a specific function exist with these properties?

lets say i have a function where: $[p,c,n,k] \in \mathbb{Z}$ defined some way in this function $ f(x) = \sum_{q=0}^{n} \sum_{v=1}^{k-q+1}\frac{c(k+1-v)!(p-k)!}{(k+1-v-q)!q!(p-k-q)!}x^{k-v+1-q} = ...
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If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \to \sqrt{x}$

If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \rightarrow \sqrt{x}$ for $x > 0$. Let $\epsilon > 0$ be arbitrary, want to find $N \in \mathbb{N}$ such that $n \geq N \Rightarrow ...
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generalization of geometric series $ \sum_{k=0}^\infty x^{p(k)} $

I have been playing around with infinite series and I wondered if it is possible to find an expression for the series: $$ \sum_{k=0}^\infty x^{p(k)} $$ as a generalization of geometric series. $p(k)$ ...
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How to calculate these limit superiors and limit inferiors?

Let $\{a_n\}$ and $\{b_n\}$ be two sequences of positive real numbers, such that $$a_{2k} = \frac{1}{3^k}, \, \, \, a_{2k-1} = \frac{1}{2^k},$$ and $$b_{2k} = \frac{1}{4^{k-1}} , \, \, \, ...
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Is a closed-form expression of this non-linear expression attainable?

A sequence is defined by: $$a_{n+1}=a_n+\frac{1}{a_n}$$ With $a_1=1$ Is a closed-form expression for this sequence possible? A search turns up the rule that all expressions with linear recurrence and ...
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80 views

Taylor Series Expansion for Function of Two Variables (with Countable Discontinuities)

Given a real-valued function of two real variables, under certain conditions of smoothness in a closed ball about some point, we can obtain a Taylor series for the function about that point. I want ...
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Name for hypergeometric-like sum.

Consider the sum $F(a_1,\cdots,a_r; z)\sum_{m=0}^{\infty}\frac{(a_1+m)(a_2+m)\cdots(a_r+m)}{(a_1)(a_2)\cdots (a_r)}\frac{z^m}{m!},$ where $a_i$ for $i=1,2,\cdots,r$ is an increasing sequence of ...
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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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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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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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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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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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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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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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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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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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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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45 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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58 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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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 ...