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

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Non-linearity on Random Sequence

I recently incorrectly assumed that applying a non-linear operation on a completely uncorrelated sequence would yield an uncorrelated sequence. Turns out that it is trivially easy to show that this ...
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49 views

Proof by induction that $\sum\limits_{n=1}^{\infty} \frac{1}{k^n} = \frac{1}{k-1}$, is my proof correct?

I proved ($\underline{see}$ Proof by induction that $\sum\limits_{k=1}^n \frac{1}{3^k}$ converges to $\frac{1}{2}$) the case for $\sum\limits_{k=1}^{\infty} \frac{1}{3^k} = 1/2$, so I thought I would ...
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48 views

Limit of infinite series involving harmonic numbers

The $k^{th}$ harmonic number is given as \begin{equation} H_{k} = \sum_{i=1}^{k} \frac{1}{i} \end{equation} I am interested in the following series: \begin{equation} \sum_{k=1}^{\infty} ...
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29 views

Will every eigenvalue in this type of matrix eventually be a common eigenvalue to infinitely many subsequent larger matrices of the same form?

Let: $$a(n) = \lim\limits_{s \rightarrow 1} \frac{\zeta(s)\sum\limits_{d|n} \mu(d)(e^{d})^{(s-1)}}{n}$$ $$a(n) = ...
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36 views

Help with whether series converges

how can you tell if $\sum_{n=1}^\infty \frac{(2n-1)!!^{1/5}}{(2n)!!^{1/5}}$ converges or not? I tried Raabe's test but didn't get anywhere. Thanks in advance.
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37 views

How to prove the double summation of series?

I find a series in the paper of Borwein, it reads $$ \sum_{(x,y)\in Z^2/(0,0)}\frac{(-1)^{x+y}}{(x^2+3y^2)^s}=-2(1+2^{1-s})\alpha(s)L_{-3}(s) $$ where $\alpha(s)$ is alternating zeta function, and ...
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28 views

3rd Roots of Unity series

I'm interested in the series representation of the following function: $$f(x)=\frac{e^{\omega_3^0x}+e^{\omega_3^1x}+e^{\omega_3^2x}}{3}$$ where $w_3^k=e^{\frac{2i\pi\cdot k}{3}},k=0,1,2.$ By Euler's ...
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39 views

Would this derivation of an integral formula be valid?

So, employing the method of synthetic division, I derived that or in much simpler form Given this, my question is, would the following statement be valid?
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32 views

Show that the sequence $\sum _{k=1}^{\infty}\frac{e^k}{k\log(k+1)}$ is convergent in $c_0$,but not absolutely convergent in $c_0$

Let $x_N = \sum_{n=1}^N $${e^k}\over { k\log(k+1)}$ since $\log(1+k)\le k$ , then we see that $\|x_N\| \le 1$ This question appears in a functional analysis paper but the function does not depend on ...
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36 views

Weird question about interval of convergence

The question is: if $$ f(x) = \sum \limits_{n=0}^\infty x^n$$ determine the interval of convergence for the power series representation of $$\int_0^x f(t) \, dt$$ That integral threw me off.
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37 views

Infinite sums over integral of triple associated Legendre polynomials

I have a couple of integrals of triple infinite sums of associated Legendre polynomials, which I'd like to integrate using Gaunt's Formula. Any help would be very much appreciated, as I'm really ...
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44 views

There exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(z)| = \infty$

Suppose $f$ has an Essential Singularity at $Z_0$. Then there exist a sequence $Z_n$ with $Z_n \to Z_0$ such that $\lim_{n \to \infty} |f(Z_n)| = \infty$ Here two cases arise If there exist a nbd ...
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29 views

Show that a sequence is between a range

I got this question in class which I'm having trouble proving I tried investigate the sequence a little bit but it doesn't seem like I'm doing the right think, some help? $ \frac{39}{e^2} \le ...
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64 views

Is the Ulam Spiral just a coincidence?

I was messing around with the Ulam spiral because I was a little skeptical on it having any actual relevance. I noticed that if you lay out the spiral and then circle all the even numbers, it displays ...
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27 views

Estimate from above $\sum_{m=1}^{n-1}\frac{1}{n^\alpha-m^\alpha}$

Find an upper bound for $$\sum_{m=1}^{n-1}\frac{1}{n^\alpha-m^\alpha}$$ with $\alpha>1$. I do not know where to start but, for example, if $\alpha=1$ the previous sum is linked to Harmonic numbers ...
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58 views

Uniformly convergence question

I have the following exercise: Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to ...
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20 views

Analytic function with alternating taylor series

Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$? It's not hard ...
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86 views

Arithmetic progression Find first term and common difference when sum of 10 terms and the 8th term is given

Sigma is a car company that sell cars. Sigma sells $x$ cars in the first month and its sales increase constantly by $y$ cars every subsequent month. It sells $96$ cars in the $8^{th}$ month and the ...
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25 views

Convergent sequences doubt from Rudin

What does the following mean: The sequence $<1/n>$ converges in $\mathbb R$ (to 0) but fails to converge in the set of all positive real numbers [with $d(x,y) = |x-y|$]. Reading Rudin's ...
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39 views

Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the ...
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75 views

How To Know in a Application Sequence/Series Problem Which Variable is $a_{0}$ or $a_{1}$?

How do you determine what value takes on $a_{0}$ or $a_{1}$ in order to use either the infinite sum formula or $a_{n}$ formula? For example, consider the following two problems: A certain culture ...
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34 views

Sequence name and properties

I was recently watching a lecture on networks and noticed an interesting real number sequence there: $$x_n = x_{n-1} + 1/x_{n-1}$$ with $x_0$ equal to some constant $c$. I started thinking about this ...
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36 views

Fourier series of cotangent

I have found the Fourier series of $\cos(ax)$ and i get: $$\frac{ \sin(a \pi)}{a\pi}\left[ \left(\frac{1}{2a^2}\right)- \sum_1^\infty \frac{(-1)^n \cos(nx)}{n^2-a^2}\right]$$ How can I deduce the ...
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21 views

Is this statement about Abelian/Tauberian theorems true?

Suppose we have some real constants $c_n \geq 0$, and know that $$\sum_{n=0}^{\infty} c_nr^n$$ converges for all $r \in (0,1)$. Suppose that the limit $$\lim_{r \uparrow 1} (1-r)\sum_{n=0}^{\infty} ...
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50 views

compact convergence for a series in complex space

I need some help with this. I have to show that the follwing series converges compat. $$\sum_{n=1}^\infty f_n :D:= \{z \in \mathbb{C} | Re(z) > 0 \} \to \mathbb{C}, f_n (z):=\frac{1}{z+n^2} $$ I ...
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49 views

$\sum a_n$ converges $\implies\ \sum \sqrt{a_na_{n+1}}$ converges?

Let $a_n > 0.$ When $\sum a_n$ converges $\sum \sqrt{a_n a_{n+1}}$ converges or not? For, $$\frac{\sqrt{a_n a_{n+1}}}{a_n}=\frac {\sqrt{a_{n+1}}} {\sqrt{a_n}}$$ $\because$ By comparison test ...
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33 views

Finding an explicit expression for a sequence

I am looking for a method (if it even exists) to find the closed form for sequences like : $u_{n+2}=\sqrt{\alpha u_{n+1}+ \beta u{n}}$ with $\alpha,\beta,u_0,u_1>0$ I already know a method exists ...
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49 views

Elementary proof about nth differences of nth powers of integer

In a post on Math.SE., a proof sketch was proposed for the proposition below: The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$. ...
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69 views

Multinomial Theorem for Negative Exponents

Using an analog to Newton's binomial theorem with negative exponents, is it true that $$ \begin{align} \left(\sum_{k=0}^mx^k\right)^{-n} & = \sum_{0\le ...
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40 views

Uncompleted Summation Series

I have run across a problem that requires me to find the sum of a "geometric" series up to the 20th iteration, starting at n=1. The series is given as follows: ...
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65 views

What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
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how to integrate this equation with two sums inside?

i'm reading a book, and i have trouble with this problem, i don't how to integrate this equation and where to begin from. Although they already give the answer but I don't understand how to get it. I ...
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37 views

A special limit-sum interversion

Let $l\in\mathbb{R}$ and $(u_k)_{k\in\mathbb{N}}$ a sequence converging to $l$. Let $a_{n,k\in\mathbb{N}}$ be such that $\displaystyle\forall n\in\mathbb{N},\sum_{k=1}^na_{n,k}=1$. $\forall ...
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26 views

Number of terms needed to estimate a series.

How many terms are required to estimate the sum of $$\sum_{n=1}^{\infty} \frac{12(-1)^{n+1}}{n^2}$$ with an error of less than .005 My first issue is whether my remainder should be .005 or .0045 ...
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28 views

Convergence of double series $\sum_{n,m\in\mathbb Z} c_n \bar{c}_m \frac{\sin(n-m)}{n-m}$.

Let $$\sum_{n,m\in\mathbb Z} c_n \bar{c}_m \frac{\sin(n-m)}{n-m}$$ where $c_n$ is a sequence of complex numbers for which $\sum_n |c_n|^2<\infty$; $\bar{c}_m$ is complex conjugate of $c_m$. Does ...
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29 views

Evaluate the value of $f_{mn}$

Let $$ f_{m,n}=\sum_{k=0}^{mn-1}(-1)^{\lfloor\frac{k}{m}\rfloor+\lfloor\frac{k}{n}\rfloor} $$. Here the $\lfloor x\rfloor $ means the floor function. Evalute the value of $f_{m,n}$. I tried use $$ ...
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47 views

Non-uniform convergence example

For the past couple of days, I've been trying to come up with a example for a problem in which $\sum_{k=1}^{\infty} |f_{k}(x)|$ does not converge uniformly but $\sum_{k=1}^{\infty} |f_{k}(x)|$ ...
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20 views

How to show the series of expectations for truncated symmetric random variables is convergent

Suppose that $(X_n)$ is i.i.d. with symmetric distribution and that $E(|X_1|)<\infty$. I want to show that $\sum\limits_{i=1}^{\infty} \frac1iE(X_i 1_{[|X_i|<i]}) $ converges. Attempt: Since ...
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51 views

Degree of a Sequence of Squares Recurrence Relation

We are told that there exists a k-th order homogeneous linear recurrence relation $a_n=r_1a_{n-1}+...+r_ka_{n-k}$ which has distinct roots. We need to prove that for every $a_n$ that is satisfied by ...
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38 views

Hilbert's double series theorem $\sum_{n,m}\frac{a_n b_m}{m+n}$ and its generalizations.

Let $l_p$ be the space of all complex sequences $x=\{x_n\}$ equipped with the finite norm $\|x\|_p=\left(\sum_n |x_n|^p\right)^{1/p}$. Hilbert's double series theorem states that $$\sum_{n,m}\frac{a_n ...
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28 views

Calculate infinite sum

I would like to find a closed for the following sum: $\sum_{i=1}^\infty a^{i^\beta}$ Where $|a|<1$, and $\beta\geq 1$. Anyone any ideas? Best, Ben
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Given $u_{n+1}=u_n+\frac{1}{n^\alpha u_n}$. Show that $u_n\leqslant \sum_{k=1}^{n}\frac{1}{k^\alpha u_1}$?

Let $u_{n+1}=u_n+\dfrac{1}{n^\alpha u_n}$ for all $n\geqslant1$ and $u_1>0$. Show that for all $n\geqslant 1$: $$\displaystyle u_n\leqslant \sum_{k=1}^{n}\dfrac{1}{k^\alpha u_1}.$$ I found that ...
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61 views

An upper bound for $\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$.

Let us consider $a_n, \phi_n, K_{n}$ complex sequences. Let $$\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$$ where $\left|K_{n}\right|\leq \gamma$ and $\gamma>0$. Can we ...
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66 views

Does a Banach valued Cauchy series over arbitrary set converges?

Definitions: If $f$ is a function from a set $A$ (not necessarily countable) into a Banach space $V$, we say that the series of $f$ over $A$ converges if there exists an element $v \in V$ such that ...
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33 views

Two statements about partial limits and intervals

Let $(a_n)$ be a sequence and $I=(a,b)$ an open interval such that every $L\in(a,b)$ is a partial limit of $(a_n)$. Decide whether or not the following statements are true: $\{a_n | n \in ...
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54 views

Weak continuous convergence of operators

Let $T_n$ and $T$ be linear maps from Banach space $X$ to a Banach space $Y$. Suppose $T_n$ satisfies $T_nx \to Tx$ (convergence in the $Y$ norm) for all $x \in X$. Let $x_n \rightharpoonup x$ in ...
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27 views

Shifting limits in series solutions to ODEs

I'm trying to practice the Frobenius method of solving ODEs, and I keep getting the answer wrong. It seems to be down to the shifting of limits of the sums, although it is not clear in the solutions I ...
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50 views

The $L^p$ convergence rate of the tail of the series $\sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} \}2^{-na}$

This a follow-up to the question: Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-ja}$ When $a > 0$, we have $$ \sum_{n=1}^{\infty}\min\{1,2^n |x|^{-1} ...
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32 views

Convergence Rate of the Tail of the Series $m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-na}$

When $a > 0$, it's fairly easy to see that $$ m^{a}\sum_{n=1}^{\infty}\min\{1,2^n m^{-1} \}2^{-na} \leq C \quad \forall m \in \mathbb{N}, $$ where $C$ is a finite constant independent of $m$. How ...
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32 views

How would I determine if this infinite series is convergent or divergent using the limit comparison test?

$$\sum_{n=1}^\infty {2^n \over3+4^n}$$ My thinking is that $4^n$ will grow much more rapidly than $2^n$, and the +3 in the denominator is negligable. Therfore, I should compare it to ...