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

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112 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+ ...
2
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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 ...
2
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80 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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124 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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193 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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78 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 ...
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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 ...
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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 ...
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72 views

To find the limit of three terms mean iteration

We know that the arithmetic-geometric mean $AGM(a,b)$ of $a$ and $b$ defined as $$2a_1=a+b$$ $$b^2_1=ab$$ $$2a_n=a_{n-1}+b_{n-1}$$ $$b^2_n=a_{n-1}b_{n-1}$$ $AGM(a,b)=\lim\limits_{n\to \infty} ...
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210 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 ...
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177 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 ...
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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 ...
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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 $x_0>0$
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58 views

Asymptotics and Related Properties

I have a rather general question: If there are two integer sequences such that $$\lim_{n\to\infty}A(n)/n=\lim_{n\to\infty}B(n)/n=c$$ is there anything else that can be said about them necessarily? ...
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605 views

cauchy sequence and necessary and sufficient condition for convergence

Question: Show that for a sequence $\{x_m\}$ of real numbers to be a Cauchy sequence, it is necessary, but not sufficient that $|x_{m+1}-x_m|$ converges to zero. This is how I proved that it is ...
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0answers
140 views

Simplify the square of a sum of cosine functions

I have a square sum of exponantials as below: $$\left|\sum_{l=0}^{M-1}\exp\left(jl^2a\right)\,\exp\left(\frac{-j2\pi l}{M}b\right)\right|^2 $$ where $a$ is constant and $b$ is an integer . and I have ...
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54 views

Least upper bound for a positive real sequence satisfying $|x_u−x_v|\cdot|u−v|>1$

The starting point of this question is the: IMO 1991, problem 6: Prove that for any $\alpha>1$, there exist a bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ such that, if $u,v$ are distinct ...
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187 views

Figuring out expression to give a integer sequence

Here given is a sequence from OEIS. The sequence is triangle of coefficients from fractional iteration of $e^x - 1$. Few terms are: $1, 1, 3, 1, 13, 18, 1, 50, 205, 180, 1, 201, 1865, 4245, 2700, 1, ...
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56 views

Representing series $f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$ as a Dirac comb function.

Consider the function $$f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$$ where $\omega_n= \sqrt{(\frac{n \pi c}{l})^2-(\frac{r_0}{2})^2}.$ If we neglect the term ...
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412 views

Fourier series for piecewise function

let $-\pi \leq x\leq \pi$ and $$f(x)=\begin{cases}-x-\pi, & \text{ if} -\pi \leq x\leq -\pi/ 2\\ \;\;\;x, & \text{ if } -\pi/2 \leq x\leq \pi/2\\ -x+\pi, & \text{ if } \pi/2 \leq x\leq ...
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66 views

Simplify series of powers

Is there any closed form for the following series? $$\sum_{m=0}^{\infty}\,\frac{1}{m!(m+1)!}\,(-\alpha^2)^m\,\, \sum_{n=0}^{\infty}\,\frac{(m+n)!}{n!(n+1)!}\,(-\beta^2)^n$$ I only know that the ...
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67 views

Simplify series of exponentials

I would like to simplify the following series: 1.$$\sum_{n=1,odd}^{\infty}\frac{\exp(-an^2)}{(b-n^2)^2}$$ 2.$$\sum_{n=2,even}^{\infty}\frac{\exp(-an^2)}{(b-n^2)^2}$$ with $a$ and $b$ $\in ...
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40 views

Growth of partial sums of a divergent series

I am trying to find how the product $\prod_{n=0}^Me_n$ grows with $M$, when $$e_n=1+\frac{a_1}{t+n}+\frac{a_2}{(t+n)^2}+\dots$$ with large $t$. So as a first estimate I took $e_n=1+\frac{a_1}{t+n}$ so ...
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71 views

Bounding an implicitly defined sequence

I have a sequence $\lambda_0,\lambda_1,\ldots,$ which is defined implicitly as $$ \lambda_0 = \frac{1}{2},$$ and $$\lambda_{k+1} = \max_{\lambda\in[1,b]} \left\{\frac{1}{2\lambda}\prod_{0\leq ...
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0answers
94 views

When $\ell^2$-convergence implies $\ell^1$- convergence?

Consider a sequence $(x_n)_{n\in\mathbb N}$ in $\ell^1$ (sequences taking their values in $\mathbb R$), where $x_n=(x_{i,n})_{i\in\mathbb N}$. What are sufficient conditions on the sequence ...
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0answers
116 views

Differentiate or integrate to find a power series?

I was just introduced to the concept of integrating/differentiating to find a power series, but I'm unsure about which method to use. Does it matter? Is it a matter of convenience?
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89 views

Closed form formula for a double series related to wave equation

Does anyone have a closed form formula for the double series $$\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)?$$ This is related ...
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60 views

Power series dense set of points of convergence

Give an example of power series with dense subset of points on a circle at which the series is convergent and dense subset of points on a circle at which it is divergent. Could you tell me how to ...
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60 views

Estimate the error due to replacing the sum of the series with its first n terms

Estimate the error due to replacing the sum of the series with the sum of the first $n$ terms. $1+ \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!}-...+\frac{(-1)^n}{n!}+...$ The hint in the textbook says ...
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62 views

Proof concerning sequence and series of functions

Let $\{s_n\}_{n=1}^{+\infty}$ be a sequence of distinct points in the interval $(a, b)$, $\{c_n\}_{n=1}^{+\infty}$ be a positive sequence such that the series $$\sum_{n=1}^{+\infty}c_n$$ converges. ...
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54 views

Distorted Newton binomial

Let $\varepsilon >0$, with $\varepsilon \neq 1$. Consider the sequence $u_n$ defined by $$ u_n=\sum_{k=0}^n \binom{n}{k}^2 (-\varepsilon)^k $$ Is is true that $(u_n)$ takes on negative and ...
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61 views

Is the answer given to Calculus 9e by Larson section 9.5 question 37.

The problem Approximate the sum of the series by using the first six terms. question #37) $\sum \limits_{n=0}^{\infty} \dfrac{\left(-1\right)^n 2}{n!}$ The answer given by CalcChat*: ...
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62 views

General weak law of large numbers, example from Chung's book.

I am trying to solve an example from the book "A Course in Probability Theory" by Chung, Third Edition page $118$. How can I prove the following asymptotics: ...
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238 views

Sequences of Natural Numbers without Arithmetic Subsequences

Let's call a sequence $k^+$-free if it is contains no arithmetic subsequence of length $k$. Define the $\bf{density}$ of a sequence of natural numbers $s_n$ as $$\lim_{n\to \infty} \frac{n}{s_n}$$ ...
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90 views

Harmonic series with some numbers deleted for what $\alpha$ is $\sum ' \frac{1}{n^{\alpha}}$.

I have a question about a subseries of harmonic series with reciprocals of natural numbers containing a certain digit deleted. I know how to prove that such series is convergent when we delete all ...
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0answers
342 views

Find a tight upper bound of the following expectation.

I am stuck in finding a tight upper bound (as tight as possible) of the following expectation $$E\left [ (1-a\cdot b^{X})^{m} \right ]$$ where $X\sim B(n-1,p)$ is a binomial random variable.In ...
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100 views

A more general exponential integral

More generalized on the previous question: A improper integral with Glaisher-Kinkelin constant Show that : ...
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110 views

Two trigonometric and exponential integral

You may view my related problem: A hard definite integral with trignometric Show that : $$\int_0^{\frac{\pi}{2}}x\sqrt{\tan x}\text{e}^{2ix}\text{d}x=\frac{\pi ...
2
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0answers
74 views

Can we simplify this result?

I was evaluating $$I=\int_0^1\ln (1+x)\ln ^{n-1}x\frac{\text{d}x}{x}$$ I did it with series: $$I=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\int_0^1x^{k-1}\ln ^{n-1}x\text{d}x$$ ...
2
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0answers
290 views

Prove that sum is finite with the help of generating function

Please help me to prove that the following sum is finite $$ \sum_{j=2l-2}^{\infty}j!\, a_j^{(l)}, $$ here the generating function of $\displaystyle{a_j^{(l)}}$ is ...
2
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0answers
131 views

Do we have closed form for these series?

Continuous on the previous question, can we get a closed form for these? $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( ...
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0answers
41 views

Compute $ \operatorname{Li}_{3}\left(\frac{1}{2} \right) $

Where could I find a proof of the identity $$ \operatorname{Li}_{3}\left(\frac{1}{2} \right) = \sum_{n=1}^{\infty}\frac{1}{2^n n^3}= \frac{1}{24} \left( 21\zeta(3)+4\ln^3 (2)-2\pi^2 \ln2\right)$$ ?