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

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56 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop (...
2
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0answers
27 views

Finding a base for a series to sum to a constant

I'd like to find the value of $r$ that solves the following equation: $$\sum_{n=1}^N r^{\frac{-1}{n}} = C \,,$$ where $N$ and $C$ are positive constants. An approximate method would also work fine ...
2
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0answers
278 views

Equality between an infinite product and an infinite series. How can I reconcile both?

Maybe a trivial question, but how could I reconcile the following equation: $$\displaystyle \prod_{n=2}^\infty \left(\frac{1}{1-\frac{1}{n^2}}\right)^{(-1)^n}=\sum_{n=1}^\infty \left(\frac{1}{(2\,n -...
2
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0answers
97 views

Product of strong and weakly converging sequences

Consider a sequence of functions $\{u_n\}\in L^2([0,T],L^2(\Omega)) $ which converges strongly to a function $u\in L^2([0,T],L^2(\Omega))$. Then $u_n \rightarrow u \;\; a.e. $ in $Q=\Omega\times[0,T]$....
2
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72 views

Random walk with $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} < \infty$

Consider a random walk started at $S_0=0$, denoted $S_n = \sum_{k=1}^{n}X_k$, where $X_1$, $X_2$... are the i.i.d increments. If we have $\sum_{n=1}^{\infty} \frac{1}{n} \mathbb{P}\{ S_n > 0 \} &...
2
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37 views

Limits of summations with different indices

In general, is it true that the limit as $n$ goes to infinity of $\sum_{i=1}^n x_{i}$ is the same as the limit as $n$ goes to infinity of $\sum_{i=1}^{n-1} x_{i}$ for some sequence $x_{i}$?
2
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137 views

How can the following “funny identity” be generalised?

When asked for a "funny identity", Andrey Rekalo answered the following: $$\left(\sum\limits_{k=1}^n k\right)^2=\sum\limits_{k=1}^nk^3 .$$ Not only do I think it's funny, I also think it's very ...
2
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83 views

Proof for $\bigwedge_{n=1}^\infty\bigvee_{k=n}^\infty x_k = \limsup x_n$

How I can show the truth of this relationship? $$\bigwedge_{n=1}^\infty\bigvee_{k=n}^\infty x_k = \limsup x_n$$ The definition of $\limsup x_n$ is $$\limsup x_n = \inf_n\left[\sup_{k\ge n}x_k\right]$...
2
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212 views

Nice convergent subsequence of $\cos(n)$.

This question is related to a few questions which have been posted on the website : Is there a limit of $\cos(n!)$ Converging subsequence on a circle The limit of $\sin(n!)$ Because of the ...
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187 views

Unordered summation

I have a question regard unordered sumation. Definitions: Let $X$ be a set (possible uncountable) and let $f: X\rightarrow \mathbb{R}$ be a function. We say that $\sum _{x\in X}f(x)$ converges ...
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612 views

Series of functions: convergence, continuity, differentiability

I'm trying to work on the function series problems and I'm confused with the necessary and sufficient conditions for a function $$f(x)=\sum_{n=1}^{\infty} f_n(x)$$ to be: well-defined continuous ...
2
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61 views

Sufficient condition for absolute convergence of series

I want to prove the following statement If $\sum_{n\in I} a_n$ converges with any rearrangements of a countable index set $I$, then $\sum_{n\in I} a_n$ is absolutely convergent. The finite case ...
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63 views

Complex integral with branch cuts.

The problem is the following, $$ \ \int_{-\infty}^{\infty} du e^{-iu w }\bigg( \cos (\theta u) - i \frac{\xi }{\theta} \sin (\theta u)\bigg)^{-1/2} $$ When we go to complex $u$ plane there are branch ...
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104 views

Inequality with Fibonacci numbers

The sequence $F_n$ of natural numbers defined by equation $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0, F_1=1$ is called the Fibonacci sequence. The n-th term in the sequence is called the n-th Fibonacci ...
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70 views

Help with Series Convergence

Can someone help me prove that the following series $S$ converges: $$S=\sum_{m=1}^\infty\frac{1}{m^2|\sin(m)|}$$ I would appreciate any help in constructing a simple proof.
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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, ...
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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: $$e^{-\alpha}\,\sum_{n=0}^{\infty}\frac{\alpha^n}{n!}\,\sum_{k=0}^n\frac{\beta^k}{k!}(-...
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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 X}f(...
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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 ...
2
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115 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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88 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 \underbrace{\exp(\exp(...
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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 $f(x)=\sqrt{...
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0answers
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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120 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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111 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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0answers
180 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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823 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 $|x_m-x_n|=\frac{1}{(n+1)^...
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0answers
117 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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0answers
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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0answers
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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0answers
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} + \frac{x-y+b\cdot\...
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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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168 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} \prod_{k=...
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0answers
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+ \...
2
votes
0answers
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 ...
2
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0answers
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,\ 0\...
2
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0answers
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 ...