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

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Examples of three integer sequences defined via indirect recursion (ie. in terms of each other)

I'm aware of the Hofstadter Male and Female sequences, which are a pair of integer sequences defined via mutual recursion: $$ M(0) = 0 $$ $$ F(0) = 1 $$ $$ M(n) = n - M(F(n-1)) $$ $$ F(n) = n ...
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211 views

variants of geometric series

My question can $\displaystyle \mathbf{\sum_{n \geq 0} a^{\lfloor n \sqrt{2}\rfloor}}$ be expressed as the sum of rational functions in a? Here $\lfloor \alpha \rfloor$ is the floor function, the ...
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666 views

Notation for a subsequence of a sequence

If we have a sequence (an ordered list) $$ S=(s_0,s_1,...,s_n). $$ What is the notation for expressing that $S'$ is a (ordered) subsequence of $S$?
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70 views

A composition on finite integral sequence

I want to know if there is a polynomial formula for this (in general): Given $f(x)=\sum _{i=0}^n a_ix^i$ where $a_i, x^i , n \in \mathbb N^* $ . Given $f_1(n) = f(x)$, we define recursively ...
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394 views

Norms that are not equivalent

Are there uncountably many not Lipschitz equivalent norms on the space of real sequences with only finitely many non-zero elements? Thanks. (If so, how might one find/construct them?) Thanks.
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79 views

Computing the PDF of a product of the sum of 2 Nakagmi-m R.V.s with a Normal R.V

I really have two questions: One is about computing a PDF and the second is about how to sum a series involving $K_v(x)$ that the PDF in question seems to contain. I have come across the following ...
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93 views

Finite Levenshtein distance?

Is there a standard term for the relation on sequences where two sequences are related iff they have a finite Levenshtein distance, or for the equivalence classes it induces?
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327 views

Applications of Convergence of a series in Algorithms

We were introduced to testing the convergence of a series & calculating the point of convergence in the first maths of college curriculum. I wish to explore its usage in computer algorithms. ...
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126 views

Certain infinite series, arising from integral, equals zero

I want to prove that the following infinite series converges to 0: $$ \sum_{i,j\ge 0} (-1)^{i+j}\frac {2i-j}{(j+1)(2i+1)(2i+j+2)} = \sum_{a,b \in N, a\equiv 1(2) } ...
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66 views

Heuristic for adding items to a sorted list when exact comparison is unknown

a sorting question! Background: I've got a list of ~500 scrum tasks with a coarse priority (very high, high, medum, low, very low) and of course unique ids. Using those, I can define an ordering ...
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147 views

Orthogonal polynomial interpolation of a function

I want to write down an arbitrary function $f$ as an (infinite) sum of orthogonal polynomials, e.g. for $f(x) = e^{\sin(x)}$, $f(x) = \sum{a_n T_n}$, where $a_n$ are the coefficients and $T_n$ are the ...
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103 views

How to interpret the sum notation on $\sum_{r:i(r)=i} \langle R_{r\alpha} \rangle^{(t)}$

While doing research for my thesis, I ran into a paper called "Statistical Models for Co-occurrence Data". In the early pages, when talking about an iterative numerical method (a custom EM-method, to ...
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139 views

Series for Digamma function

This problem arose in finding a sum of expectations of Geometric RVs. Any idea how to simplify the sum $$\sum_{k=0}^{\frac{n}{2}-1}\frac{1}{(2n^2)^{c'}-(n^2+4k^2)^{c'}}$$ where $c'$ is ...
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108 views

Is that series-transformation known in the context of divergent summation

Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the ...
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29 views

Proof of convergence of Kaprekar's Constant

I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact. Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' ...
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13 views

proving a fraction with 2 parameters to be small

Hi I have a fraction as below $$\frac{1.623x^4+0.434x^4\sum_iy_iz_i^2+(0.014x^2+0.0027)\sum_iy_iz_i^4}{1.645x^2+(0.083-0.329x^2+0.435x^4)\sum_iy_iz_i^2+0.014\sum_iy_iz_i^4}$$ where $x\in[0, 0.5]$, ...
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61 views

How prove $\frac{1}{\sqrt{2006}}<a_{2006}<\frac{2}{\sqrt{2006}}$ in sequence?

Let $\{a_{n}\}_{n=0}$ be defined recursively by $a_{0}=\pi/4$, and $a_{n+1}=\sin a_{n}$ for $n=0,1,\ldots$. How prove $\frac{1}{\sqrt{2006}}<a_{2006}<\frac{2}{\sqrt{2006}}$?
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31 views

Convergence of a series using limit test

As per my understanding: When we take $\lim_{n \rightarrow \infty}$ of a function, it should approach a finite number, it converge. And if the opposite is true, it diverges. Now, the test for ...
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47 views

Another proof of the Dirichlet's test

My teacher said, that the Dirichlet's test was equivalent to the lemma as follows, and the lemma could be proved with an estimate without using Abel's summation formula. He expected me to complete the ...
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37 views

Negation - Cauchy sequence

Suppose that $(x_i)_{i \geq 1}$ is not a Cauchy sequence of real numbers. How to prove that there exist $\varepsilon >0 $ and an increasing sequence $(i_n)$ of indices such that $$ ...
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97 views

Is there some fast and efficient way for solving $x$

Let $b$ be a given constant scalar between 0 and 1, and $A$ a given $N \times N$ transitional probability matrix (i.e., each row sum of $A$ is 1, and $0\le {A}_{(i,j)} \le 1$). Let $A\circ A$ denote ...
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52 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$

Let $x_n$ be a sequence of real numbers such that $x_n\in(0,1).$ Find the sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$ My answer is $\frac{(x_n)'}{(1-x_n)^2}.$ But the term $(x_n)'$ make me ...
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12 views

How to prove a function $f(t)=\sum_{n=0}^{\infty} a_n t^{2n}$ with real coefficients $a_n$ is regular in the strip $-\pi/4\le \Im(t) \le \pi/4$?

How to prove a function $f(t)=\sum_{n=0}^{\infty} a_n t^{2n}$ with real coefficients $a_n$ is regular in the strip $-\pi/2\le \Im(t) \le \pi/2$? What will be the sufficient conditions on the real ...
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12 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
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24 views

Two case about convergent series

Could you help me to prove analytically that ? I started to study Taylor Series and I'm lost. Thanks in advance.
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19 views

Sum of squares of series of boolean variables

I am going to simplify the following series: $$\sum^4_{v=1} \left(1 - \sum^4_{i=1} x_{v,i}\right)^2 + \sum^4_{i=1} \left(1 - \sum^4_{v=1} x_{v,i}\right)^2$$ Since $x_{i,j}$ is a boolean variable, ...
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19 views

Proving that a sequence fits a certain formation rule.

I've got this function here: $$F(m):=-\frac{\sum _{i=1}^{m-1} \text{F}(i) \left(\binom{m}{i-1}-3^{m-i}\right)}{m-1}$$ $$F(1):=x$$ and if I calculate the values for $m = {1,2,3,4,\ldots,n}$ I get ...
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17 views

Limit of a sum over an increasing finite set: dominated convergence / riemann integral

Let $(S_k)_{k\in\mathbb{N}}$ be a sequence of finite sets where $S_k \subset [0,1]$ for all $k$. It is assumed that $S_k\subset S_{k+1}$ and that $$\lim_{k\to\infty }\max_{s\in S_k} (s-s')=0$$ where, ...
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44 views

How can I cleverly use the reverse triangle inequality in this case?

Say, we have the following recurrence relation: $$x_{k+2}=4x_{k+1}+x_k(-3-2h\lambda)$$ with $x_0 $ given, $x_1=(1+h\lambda)x_0, h$ small (step size), and $\lambda<0$. Here is the context is case ...
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22 views

Sum of analytic functions

The sum of analytic functions is analytic. Does it mean that: $\sum_{i}\sum_{n=0}^{\infty}a_{in}x_{i}^{n} = \sum_{n=0}^{\infty}a_{n}x^{n}$ ? Is this also true $\sum_{i}\sum_{n=0}^{m}a_{in}x_{i}^{n} ...
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30 views

Standard results for limit of recurrence relation?

Is there a standard result out there that gives the following? It looks graphically like it must be true, but I'd like to appeal to a known result if possible. Let $f:[0,1]\to[0,1]$ be a continuous, ...
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19 views

mean square convergence vs almost sure convergence

I saw a few examples that show that almost sure convergence doesn't imply convergence in mean square. Can anyone find an example of a random series that converges in mean square but doesn't converge ...
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53 views

a completion of $c_{00}$ under some norm

Let $X$ be the completion of $c_{00}$ under some norm $\|\cdot\|_X$ satisfying $\|e_n\|_X=1$ for all $n$, where we let $(e_n)$ denote the canonical unit vectors in $c_{00}$. In Albiac/Kalton's book ...
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14 views

insert parentheses to series question

i want to prove that if i have convergent series $\sum _{n=0}^{\infty }\:a_n$, so $\sum _{n=0}^{\infty }\:a_n$ $\rightarrow S$, any series that becomes from inserting brackets to $\sum _{n=0}^{\infty ...
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48 views

A general question on positive integer sequence of a certain formula

Let $A=\{\ $ a certain polynomial | all variables$\ \in\mathbb N\ \},\ A\ \subseteq\ \mathbb Z^+,\ $ such as $A = \{2n−1\ |\ n\in\mathbb N\}$. Let $B=\mathbb Z_{\ge 0}-A$. Let $C$ be the set that ...
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27 views

Help in finding a book

That is my problem: i have this series: $\sum_{k=0}^{\infty} \psi_{k}(h)\cos(k\theta)$. I need to see to what it converges assuming something on the functions $\psi_{k}(h)$. There is a book or a site ...
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23 views

How to find out convergence of sequence of functions in short time.

I was studying about sequence of functions and about uniform and pointwise convergence of these series. To prove or disprove the convergence of sequence of functions we have several methods like by ...
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31 views

series functions of complex variable $ z $ and alternating zeta function convergence

Let $f_n(z)$ and $p_n(z)$ two series functions of complex variable $z$ defined as the following: $f_n(z)$=$ \sum_{n}exp({(-1)}^{n-1}{n^{-z}})) $ $p_n(z)$=$exp(\sum_{n}({(-1)}^{n-1}{n^{-z}})$ ...
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17 views

How to characteristize or distinguish those lacunary series with same natural boundary from each other

There are lots of different lacunary series with integral coefficients and with same natural boundary. As we know,Some functions can be distinguished or characterized by their poles. Now,the ...
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18 views

Show that if $\lim_{k \rightarrow \infty} \sup (\sqrt[k]{|a_k|}) < 1$ then $\sum_{k=0}^\infty a_k$ converges absolutely.

Let $(a_k)_{k \in \mathbb{N}}$ be some real sequence. Show that if $\lim_{k \rightarrow \infty} \sup (\sqrt[k]{|a_k|}) < 1$ then $\sum_{k=0}^\infty a_k$ converges absolutely. I have some general ...
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32 views

Step in proof involving Euler-Mascheroni constant

I was just looking at a proof that shows how the Euler-Mascheroni constant exists and is situated between 0 and 1 . However, I stumbled across a step in the proof that doesn't seem very obvious to me ...
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40 views

Simple Math Problem on Interval

It's not clear for me. I see this wikipedia page for a difference of half interval on $\mathbb{R}$ and interval on $\mathbb{R}$? For example $$ \{ (-\infty \le x \le a) \, \left|\, a \in \mathbb{R} ...
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27 views

geometric convergence of sequence

Let $\{x_n\}$ be a sequence of non-negative real numbers. For fixed $m \in \mathbb{N}$ Denote $$x_k^{(m)}:={1 \over m}\sum_{i=1}^m x_{k+1-i},\forall k\geq m$$ Assume $x^{(m)}_k\leq ...
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42 views

Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
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22 views

Finding coefficient $c_n$ of Cauchy product for $a_k = (\frac{-1}{2})^k, b_k = \frac{(-1)^k}{k+1}$

I need to find the coefficient $c_n=\sum_{k=0}^n a_k b_{n-k}$ for the Cauchy product $\sum_{k=0}^\infty c_k=(\sum_{k=0}^\infty a_k)(\sum_{k=0}^n b_k)$ with $(a_k)=(\frac{-1}{2})^k$ and $(b_k) = ...
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56 views

off by one when trying to solve the sum of a geometric series

$$\frac{1}{3} - \frac{1}{9}+\frac{1}{27}+\cdots+\frac{1}{3}\left(-\frac{1}{3}\right)^{n+1}+\cdots$$ If I use the formula $S=\dfrac{a}{1-r}$ where the first term in the sequence is $a=1/3,$ and the ...
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14 views

action of transition operator on function

Let $P$ be the transition operator of a markov chain with discrete time and discrete state space $X$. The action of the transition operator on a function $X \to \mathbb{R}$ is defined by $Pf(x) = ...
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15 views

How to determinate the convergence interval D'Alembert

I need to determinate the interval of convergence using the generalized criterion of D'Alembert. Considering $\sum\limits_{n=1}^\infty nx^{n-1}$ To determinate the interval of convergence, I did : ...
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29 views

Check divergence and convergence of series with criterions

I have 4 series and need to check if it diverge or converge. Before starting, I wanted to determinate the best criterion to determinate for each of them and arrived with this : $\sum\limits_{i = ...
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102 views

Find all the subsequences which are arithmetic progression

Given a sequence is there a linear or sub-linear algorithm to find all the sub-sequences that are arithmetic progressions with a given D, where D is the consecutive difference between the elements ?