### proof writing - If $x_k ≥ 0\;\forall \in \mathbb N$, and $y_k$ a bounded sequence, then the series $\sum_{k=1}^\infty x_ky_k$ converges

Hi I'm really struggling with this proof. For a start I'm struggling to believe it's true:For example, if we take $x_k = \dfrac{1}{k^2}$ and $y_k = -k^3$ (which is bounded above by any positive number), then the series $x_ky_k$ does not converge? What am I doing wrong? I feel like I'm being insanely stupid.Thank you!...Read more

### optimization - The longest sequence of numbers with a certain divisibility property

EDIT- result. westzynthius(1931) showed that we can create a $p_x$ denizen longer than $p_x \times log(log(log(p_x)))$... Meaning for large enough prime numbers, the maximum denizen is much larger than $2p_x, 3p_x$ etc.Definition - Denizen A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P}$; and it satisfies the following condition; if "$a_{x_1} =y_1$", "$a_{x_2} =y_2$", "$x_1 \pm m_1y_1 \neq x_2 \pm m_2 y_2$ when $y_2<y_1$" and "$m_3$ isn't divisible by $y_1$...Read more

### summation - Sum of a series close to geometric

I'm stuck in finding the sum:\begin{align}S = \sum_{k=2}^n \frac{1}{2^k-1}\end{align}It seems to be close to geometric without being geometric. I wanted to try to transform it to a geometric series but I don't seem to come up with a valid transformation to do it. Anyone has an idea?...Read more

### self learning - necessary and sufficient condition for convergence of power series

Show that $\sum_{n=1}^{\infty} a^{\ln~n}$ is convergent if and only if $0<a<\frac{1}{e}$.Proof:$\ln~n<n$ for all $n \geq 1$.Hence the necessary condition for the series to be convergent will be $\lim_{n \rightarrow \infty} a^{\ln~n}=0$.And this will happen if $0<a<1$.Please give me some hint about how to proceed further....Read more

### convergence - (Dis-)proving the series $\sum\limits_n\left( 1+ \frac{1}{n} \right)^n$ converges

I am trying to prove that the series:$$\sum^\infty_{n=1}\left( 1+ \frac{1}{n} \right)^n$$converges.Now I know that$$\lim_{n\rightarrow\infty} \left( 1+ \frac{1}{n} \right)^n=e$$But how can I use that knowledge to prove the convergance ?Intuitively I would say that the series diverges since it doesn't approach zero but how can I formally prove this?...Read more

### Limit of series with exponent

I want to calculate the limit of the following series:$$\sum^{\infty}_{k=0} \frac{(-3)^k +5}{4^k}$$My first step would be to split the term into these parts:$\sum^{\infty}_{k=0} \frac{(-3)^k}{4^k}$ $\sum^{\infty}_{k=0} \frac{5}{4^k}$If both of them have a limit I can just add them together, right ?I have looked through my notes on limits and convergence but I dont know how to get rid of the exponent so I can determine the limit.I have used various online calculators but I could not understand their result....Read more

### summation - How to explain the formula for the sum of a geometric series without calculus?

How to explain to a middle-school student the notion of a geometric series without any calculus (i.e. limits)? For example I want to convince my student that$$1 + \frac{1}{4} + \frac{1}{4^2} + \ldots + \frac{1}{4^n} = \frac{1 - (\frac{1}{4})^{n+1} }{ 1 - \frac{1}{4}}$$at $n \to \infty$ gives 4/3?...Read more

### soft question - Interesting explicit convergent subsequence for not converging bounded sequence

To illustrate the (power of) Bolzano-Weierstrass theorem I am searching for an example of a bounded but not convergent sequence and an explicit convergent subsequence. I would like it to be non trivial in the sense that (cyclic) sequences like $(-1)^n$ or $1,2,3, 1,2,3, 1,2,3$ or $1,2,3,2,1,2,3,2,1$, where one can easily "see" the (constant) convergent subsequence don't count.I wanted to use $\sin(n)$, but the construction of a convergent subsequence isn't very explicit....Read more

### sequences and series - Which one is the correct notation?

I know this notation is correct:$a_1,a_2, a_3,\cdots,a_n=\left\{a_k\right\}_{k=1}^n$Now, we have a function $f(n)$.I want to write this sequence in correct notation:$\left\{ f(1),f(2),f(3),\cdots ,f(n)\right\}$ I have two notations. Which one is correct? $\bigcup_{k=1}^nf(k)=\left\{ f(1),f(2),f(3),\cdots , f(n)\right\}$ $\left\{f(k)\right\}_{k=1}^n=\left\{ f(1),f(2),f(3),\cdots, f(n)\right\}$Thank you....Read more

### sequences and series - Are these statements always true?

I haven't found an answer in my books. Although the question seems very simple, I want to ask.Are these statements always true? a) For any infinity non-negative integer sequence, if there is an exist $n-$th term closed form expression formula, for this sequence, we have always a recurrence formula. b) For any infinity non-negative integer sequence, if there is an exist recurrence formula,for this sequence, we have always $n-$th term closed form expression formula. c) For any infinity non-negative integer sequence, if there is not an exi...Read more

### sequences and series - Can the entropy of a random variable with countably many outcomes be infinite?

Consider a random variable $X$ taking values over $\mathbb{N}$. Let $\mathbb{P}(X = i) = p_i$ for $i \in \mathbb{N}$. The entropy of $X$ is defined by$$H(X) = \sum_i -p_i \log p_i.$$Is it possible for $H(X)$ to be infinite?...Read more

### sequences and series - Extension of the Jacobi triple product identity

The Jacobi triple product and the mathematical identity of it is:$$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2}$$I would like to extend the idea for $\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} h^{n^3}$My idea is below for extension:Let's assume we define $G(z,q,h)$ as$$G(z,q,h)\prod\limits_{n=1}^{ \infty }(1+zq^{2n-1}h^{3n^2-3n+1})(1+z^{-1}q^{2n-1}h^{-3n^2+3n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} h^{n^3}$$$z=ZQ^{2}h^{3}$$q=Qh^{3}$$$G(ZQ^{2}h^{3},Qh...Read more

### sequences and series - Angle between two given vector is small. Can we permute coordinates of them such that new vectors be orthogonal?

Let $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^n$ be two unit vectors such that $\sum_{i}{x_i}=\sum_{i}{y_i}=0$ $$x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n} \gt 1-\frac{1}{n} .$$ Can we prove that$$x_{1}y_{\sigma(1)}+x_{2}y_{\sigma(2)}+\cdots +x_{n}y_{\sigma(n)} \neq 0$$ for all permutations $\sigma$?...Read more

### Adding the harmonic sequence and a permutation of it

Let $\pi:\mathbb{N}\to\mathbb{N}$ be a bijection. Then does there exist another bijection $\nu:\mathbb{N}\to\mathbb{N}$ and a constant $C$ such that $$\frac{1}{n} + \frac{1}{\pi(n)} \leq \frac{C}{\nu(n)}$$ for all $n$? If so, can the constant be chosen independent of $\pi$?While the harmonic sequence $(\frac{1}{n})_{n\in\mathbb{N}}$ is what comes up in my application, I imagine that a good answer will be able to make a much more general statement about a suitable class of sequences. But I'd be perfectly happy with an answer to the question ab...Read more

### sequences and series - recursive to standard convertion

I have been trying to find an equation for a sequence, and got interested on how to convert any recursive sequence ex: $F_n=F_{n-1}+F_{n-2},\space F_0=1,\space F_1=1$ into a standard equationex: $F_n=\frac 1{\sqrt 5}(\frac {1+\sqrt 5}2)^{n+1}-\frac 1{\sqrt 5}(\frac {1-\sqrt 5}2)^{n+1}$ I decided to search around but only got beginners algebra stuff, in fact the only helpful thing I found was a video on how to do it with the Fibonacci sequence which doesn't help me with the equation I have. if anyone could give me a link to something helpful, th...Read more