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

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60 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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53 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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26 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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49 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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31 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 ...
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49 views

series of powers of integer powers

Given two real positive numbers $a,b\in(0,\infty)$ and a series of natural integers $n=1,2,3,\dots$, is there any known formula to apply in order to calculate the series $$s(n)=a^{b^n}?$$ My goal is ...
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45 views

Strange sequence needed

There's no easy way to explain this, but please bear with me. I'll try to keep it slow and simple. I'm looking for a property that is related to the generalised pentagonal numbers (A001318 in the ...
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65 views

Sum of geometric series

Let's say I have the series: $1+(x+1)+(x+1)^2….$ if $|x+1|<1$, what is the sum of infinite geometric series? This is my thinking: I have the formula $S= a \dfrac{1-r^n}{1-r}$ Now we know that ...
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107 views

Combinatorics : number of non-decreasing series of r distinct numbers where the size of series ranges from 1 to N

I understand that number of non decreasing sequences of size M with N distinct numbers is (N+M−1)C(M). However, I'm interested in finding out the number of such series of r distinct integers where the ...
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50 views

Frequency of non-increasing and non-decreasing subsequences in Matlab

Having a sequence of numbers of length L, I need to count how many non-decreasing and non-increasing sub-sequences of exact length are there. For example, if I have a sequence of length 15 $$2, 4, ...
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28 views

Why the generalized root test for series does not use $\liminf$ to get rid of the inconclusive part?

I wonder for what reasons the generalized root test for series (Roughly: $\limsup a_{n}^{1/n} < 1$ implies convergence; $\limsup a_{n}^{1/n} > 1$ implies divergence; $\limsup a_{n}^{1/n} = 1$ ...
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68 views

Closed form of generating function $r^a$

Find the closed form of the generating function of $r^a$. in this question $r$ is the variable part and $r$ assumes the values $1,2,3,4,5,6,7 \ldots$ and $a \in \mathbb R_{\geq 0}$. I would appreciate ...
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44 views

two convergent sequences

Let $a>0$. Be $(a_n)$ a sequence of real numbers . Define $x_n$ as a recurrent sequence : $$x_{n+1}=(1-\dfrac{a}{n})x_n+\dfrac{a_n}{n}$$ Prove that $x_n$ converges to $0$ if and only if sequence ...
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63 views

Sum of Power of Two Fibonacci reciprocals

Evaluate $$\sum_{k=0}^{\infty} \frac{1}{F_{2^n}} \;.$$ I'm thinking of using a relation from a term to another.
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23 views

Rate of convergence when substituting sequence by its limit

Let $(z_n)_{n \in \mathbb{N}}, (w_n)_{n \in \mathbb{N}}, (w_n')_{n \in \mathbb{N}}$ sequences of complex numbers with $|z_n| = 1$ for all $n \in \mathbb{N}$. Assume that $z_n \to z$ and $w_n \to w, ...
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46 views

Prove/disprove that if $(n a_n)$ is bounded then $(a_n)$ converges to zero

I need to prove or disprove the following statement: If $(n a_n)$ is a bounded sequence, then $\lim_{n\to\infty}(a_n)=0$ I think the statement is true but I'm not sure about my proof: Because ...
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41 views

Q Pochammer Symbol Product Identities

Consider the expression $$G(x,a) = \frac{1}{((1-a)x;a)_{\infty}}$$ Based on: Infinite sum involving ascending powers It follows that in the limit as $a \rightarrow 1$ ...
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33 views

Rearrangement of absolutely convergent series: proof verification

There is a theorem saying that if $\sum_{j=1}^{\infty}x_j$ converges absolutely, then any rearrangement $\{x_{\pi(j)}\}_{j=1}^{\infty}$ of the original series $\{x_j\}_{j=1}^{\infty}$ also converges ...
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60 views

Show that $\pi \cot \pi z = \frac{1}{z} + \sum\limits_{n=1}^{\infty} \frac{2 z}{z^2 - n^2}$

This question has been asked before, but we are supposed to prove for any $z \in \mathbb{C} \setminus \mathbb{Z}$ that $$\pi \cot \pi z = \frac{1}{z} + \sum\limits_{n=1}^{\infty} \frac{2 z}{z^2 - ...
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34 views

Polynomials satisfying Similar Appell Sequence Properties

Appell sequences, $\{P_k(x)\}$ are sequences of polynomials that have the following characteristics. $1. P_0(x)=1$ $2. P'_n(x)=nP_{n-1}(x)$ Now I have found a few sequences of polynomials that ...
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30 views

Definition - limit of a sequence - uses “rank”?

I have the following definition in my book, and was confused as to the context of the word "rank" here. The definition is as follows: A sequence $(u_n)_{n∈N}$ has limit $l ∈ R$ as $n → ∞$ (we also ...
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63 views

Is $\sum_{k=2}^{\infty}{\sin(k!)}/{(k\log(k))}$ convergent or divergent? How can I prove it?

How can I prove that $~\displaystyle\sum_{k=2}^{\infty}\frac{\sin(k!)}{k\log(k)}~$ converges ? Both Leibniz's criterion and Dirichlet's test seem rather inadequate for handling this particular task. ...
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35 views

Sums of infinite series containing terms involving Gamma function

I am looking for relevant literature on sums of series involving the Gamma function. In particular, my interest is in series of the following form. \begin{equation} \sum_{j=0}^\infty ...
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22 views

How to show $\mathbb{R^2}$ is sequentially connected without path-connectedness

Definitions: Connected: Not separated Separated: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and ...
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32 views

Using IMVT? Limit of the sequence given by the integral

I found this problem in a past exam and solve it. I want to know if I did it correctly because I used the mean value theorem for integrals (IMVT) and I'm not sure if it went the right way. Let ...
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20 views

Recurring Folds Through A Circle

If I were to cut a circular disk of paper, I would have a disk with one section. If I were then to fold it through its centre, I would have a disk with two sections (divided by the crease). If I were ...
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70 views

Does the following limit exist? (involving harmonic numbers)

Let $H_m$ denote the $m$-th harmonic number (with the convention $H_0:=0$). Fix an integer $n$. Define for $k=0,1,\dots,n-1$ $$ d_{n,k}:={1\over{n^2}}\biggl\{\sum_{j=0}^k ...
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246 views

How to prove this complex Inequality

$$\left|\sum_{i=0}^\infty\gamma^ir_i-\left(\sum_{i=0}^{n-1}\gamma^ir_i+{\gamma^nr_n \over 1-\gamma}\right)\right|\ge\left|\sum_{i=0}^\infty\gamma^ir_i-\left(\sum_{i=0}^{k-1}\gamma^ir_i+{\gamma^kr_k ...
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143 views

sum of exponential series with power increasing by geometric series

Is there any way to reduce the following summation... $2^{ar^0}+2^{ar^1}....+ 2^{ar^n} $ to a simple equation? I feel like I can pull out a $2^a$ somehow and then treat it as a normal series but I ...
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32 views

Bernoulli Number analog using Cosine (part 2)

Earlier today I posted this inquiry about the function below: $$\frac{x^2}{\cos{x}-1}=\sum_{n=0}^{\infty}\frac{C_n}{n!}x^{2n}$$ I got some good feedback but as I was playing around, I wondered if ...
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53 views

Putnam 2000 A1 Series square problem

Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which ...
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38 views

Truncated Binomial Series

Can the truncated binomial series be expressed as a closed form \begin{align} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \binom{n}{k} x^{k} \end{align}
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47 views

Evaluate $\sum\limits_{i=0}^{n}\frac{1}{6^i}$

I'm asked to evaluate: $$\sum\limits_{i=0}^{n}\frac{1}{6^i}$$ Since it's a geometric series, I got $\dfrac{1-\frac{1}{6^n}}{\frac56}$ but I think it could be wrong.
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36 views

Radius of convergence of a power series whose coeffecients are “discontinuous”

I have a power series: $s(x)=\sum_0^\infty a_n x^n$ with $a_n= \begin{cases} 1, & \text{if $n$ is a square number} \\ 0, & \text{otherwise} \end{cases}$ What is the radius of convergence ...
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On the existence of certain Weierstrass-type Extreme Value Theorems

It is well-known that Weierstrass Extreme Value Theorem can be generalized to lower and upper semi-continuous functions. Are there any other generalizations of this important result; and more in ...
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36 views

Find the radius of convergence given a series

Question: Find the radius of convergence of $\sum_{k = 0}^\infty 3^{k^2}x^{k^2} $. Attempt: By the root test $\lim_{k\to\infty} |3^{k^2} x^{k^2}|^{1/k} = \lim_{k\to\infty}|3^k x^k| = ...
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63 views

For which values of $r$ the sequence $nr^n$ converges?

We have the sequence $$nr^n$$ and we want to find the values of $r$ such that $nr^n$ converges. What I did: for $|r|<1$ we have $r = \frac{1}{k}$ where $k>1$ or $k<-1$ so: ...
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33 views

Prove that $X_n=\frac{1}{n}$ is not a contractive sequence.

So I was trying to prove this using a contradiction. I started by stating the definition of contractive which is that there exist r in (0,1) such that for all n in N, |$X_{n+2}$-$X_{n+1}$|$\leq$ ...
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68 views

Prove that if $\lim_{n\to \infty} a_n=\infty$ then $\lim_{n\to \infty} \frac{1}{a_n}=0$

let $C= \frac{1}{\epsilon}$ There $\exists N\in\mathbb N$ such that for every $n>N$, it is true that: $$a_n>\frac{1}{\epsilon}$$ We should prove that for every $\epsilon>0$ there exists ...
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41 views

How to manipulate this summation in the easiest way possible?

$$ D = \sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j}(g(x))[\ln(g) f'(x) f_c^{(j)} X_{n,k(f\rightarrow g)^c} + f_{c}^{(j)} X_{n,k(f \rightarrow g)^{c}}' + \frac{d}{dx}[f_c^{(j)}] X_{n,k(f ...
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36 views

Computation of a two variable combinatorial function

For every $t\geq1$ and $1\leq k\leq t-1$, $G(t,k)$ is a combinatorial sum satisfying the following recurrence: $$G(t,k) = 2t-1 + \sum_{h=1}^{t-k-1} G(t-k,h)\, (2k+2h-1) \;.$$ Is it possible to ...
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39 views

How to find connection between sequence and formula?

I have one sequence, which is defined like this: 4,7,10,12,15,17,20,22,24 I also found it at OEIS. And there they have a formula, which I can use to find ...
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43 views

Show the series converges uniformly

Let $f(x) = \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt n} \arctan\left(\frac{x}{\sqrt n}\right)$. Show that $f(x)$ converges uniformly. First, it is easy to see that the series converges for every $x$ by ...
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43 views

Expression for polynomial

I wonder if it is possible to find a closed form expression for following sequence: \begin{equation*} C_1=1 \end{equation*} \begin{equation*} C_2=x^2+\frac 32 \end{equation*} \begin{equation*} ...
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35 views

why $\frac{f_n}{f_kf_{n-k}}$ is an integer for this sequence

New Zealand 2013 TST problem: let $r$ and $s$ be postive integers,and Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that ...
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54 views

Question about a convergent sequence and inequality

Suppose $0 < x < 1 $ and let $n > 0 $ such that $\frac{1}{n+1} \leq x < \frac{1}{n}$. If there exists a sequence $x_n \downarrow x$, then there exists $N$ such that $\frac{1}{n+1} \leq x_n ...
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34 views

Re-arrangement of a series

If we define a series as $\sum_{k=1}^\infty s_k$, then a series $\sum_{k=1}^\infty t_k$ is a re-arrangement of $s_k$ if there is a 1-1 and onto function $f : \mathbb{N} \to \mathbb{N}$ such that ...