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

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55 views

Proving that a set of metric space is dense in $A$ iff there exists a sequence converging to $x\in A$

I'm using the following definition: A set $M$ of a metric space $(\frak M,\rho)$ is called dense in a set $A\subset\frak M$ if $$\forall \varepsilon>0,x\in A\exists y\in ...
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1answer
30 views

The sum-of-squares of the increments of an interval's partition

Let $t \in \left(0,\infty\right)$ and, for every $n \in \mathbb{N}_1$, let $m_n \in \mathbb{N}_1$ and let $t^{(n)}_0, t^{(n)}_1, \dots, t^{(n)}_{m_n} \in \left[0, t\right]$ be such that $0 = t^{(n)}_0 ...
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1answer
40 views

How does one show that the series converges almost surely?

Let $X_1, X_2, \ldots:\Omega\to \mathbb R$ be random variables. Define $C:=\{ \omega \ | \ \sum X_n(\omega) \text{ converges} \}$. There is such $q\in(0,1)$ that for all $n\in \mathbb N: P\{ |X_n| ...
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1answer
25 views

Integrating Trigonometric Series

Let $f(x)=c_0+c_1e^{i\theta}+c_2e^{2i\theta}+...+c_ne^{ni\theta}$ where $c_k\in \mathbb R$. We need to show $$\int_{0}^{2\pi}f(e^{i\theta})\overline {f(e^{i\theta})} d\theta =2\pi\sum_{k=0}^{n} c_k$$ ...
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0answers
80 views

solving a sequence problem [duplicate]

$ a_n = (1+ \frac{1} {n}) ^n $ where n is a natural number then which one is greater $ a_{2013} $ or $ a_{2014} ?$ I tried using binomial expansion but could not solve it. I also tried using ...
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3answers
58 views

What is the sum of these numbers?

I have this series: $$A=1×\frac{n}{2^0}+2×\frac{n}{2^1}+...+k×\frac{n}{2^{k-1}}$$ How can I calculate $A$? I know that the answer must be $2n$. But I do not remember how I did it then! Thanks.
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2answers
104 views

Prove that $f_n$ converges uniformly on $[a,b]$

Let $f_n$ be a sequence of functions defined on $[a,b]$. Suppose that for every $c \in [a,b]$, there exist an interval around $c$ in which $f_n$ converges uniformly. Prove that $f_n$ converges ...
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1answer
28 views

Find the value of $n$ where

Let $f(x+y)=f(x)f(y)$ where $x,y$ are natural no.s, and $f(1)=3$ Then find $n$ if $\sum_1^nf(n)=120$
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33 views

Why is this wrong? Strong Cauchy criterion.

Proposition: Suppose $(a_n)$ is a sequence such that $$|a_{n +1} - a_n| < Kr^n$$ for every $n$, then $(a_n)$ is Cauchy where $|r| < 1$ and $K > 0$ Then what follows is usually a ...
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1answer
41 views

two natural integer sequences

Have the integer sequences $f$ and $g$ defined below already been studied? Do they have any interesting property? Define the maps $f,g:\mathbb Z_{ > 0}\to\mathbb Z_{ > 0}$ as follows: Define ...
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2answers
32 views

How prove for any positive integer numbers $n$ can define Sum of the some different terms of that sequence

let positive integer sequence $\{a_{n}\}$,and such $$a_{1}=1,a_{k}\le 1+\sum_{i=1}^{k-1}a_{i},k\in N^{+},k\ge 2$$ show that: for any positive $n$ ,then $n$ can define Sum of the some different ...
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2answers
80 views

How prove this series $\sum_{n=1}^{\infty}\frac{e^n\cdot n!}{n^n}$ divergent

show that this series $$\sum_{n=1}^{\infty}\dfrac{e^n\cdot n!}{n^n}$$ divergent My try: since $$u_{n}=\dfrac{e^n\cdot n!}{n^n},\Longrightarrow ...
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0answers
24 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 ...
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1answer
45 views

Bounding sequence for $\left(1-\frac{n}{k}\right)^{\left\lfloor n/k\right\rfloor},\quad k,n\in\mathbb{N}.\quad k$ fixed

could you please help me with finding an upper-bounding sequence for the sequence $$ x_n=\left(1-\frac{k}{n}\right)^{\left\lfloor n/k\right\rfloor} $$ where $n\in\mathbb{N}$ and $k\in\mathbb{N}$ is ...
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3answers
52 views

show that the sequence ${a_n}$ is a cauchy sequence

Show that any sequence $\{a_n\}$ that has the property $|a_{n+1} -a_n|< b^n$ for $b<1$ is a Cauchy sequence. I'm having problems giving a formal proof of why this holds.
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2answers
79 views

Using Lagrange form of the remainder with cosh

I am trying to find "$\cosh 4$ using the sixth partial sum ($n=5$) of the Maclaurin series" for the function. I am also trying to use "the Lagrange form of the remainder to estimate the number of ...
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2answers
769 views

Find the value of the sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\!\Big)^{n}\right\}$

How can we find the exact value of the infinite sum $\displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\big(1+\frac1n\big)^n\right\}$? This problem appears in: T. Andreescu, T. Radulescu & V. ...
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0answers
41 views

Difference between sum of alternate terms of a decreasing series. [closed]

I'm given a series say $5,7,8,4,2,10$ First arrange terms is decreasing order i.e $10,8,7,5,4,2$. Then calculate sum of numbers at alternate positions ie $s_1=10+7+4$ $s_2=8+5+2$ Required answer ...
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1answer
202 views

Sequences, sets and element position in the set.

I have a sequence Q with the length of N. This is the fragment of this sequence: 68 70 72 74 76 78 80 The sequence has been divided into the sets of 4 elements ...
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2answers
92 views

Calculate sum of a number sequence

In a sequence of integers, $A(n)=A(n-1)-A(n-2)$, where $A(n)$ is the $n$th term in the sequence, $n$ is an integer and $n\ge3$,$A(1)=1$,$A(2)=1$, calculate $S(1000)$, where $S(1000)$ is the sum of the ...
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1answer
52 views

Catastrophic Cancellation

I need some informations about algorithms to solve the problem of Catastrophic Cancellation, or in general to calculate mean and variance of a data stream. For example about Donald Knuth's algorithm ...
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2answers
50 views

Show that there is a unique sequence of positive integers $(a_n)$ satisfying $a_1=1,a_2=2,a_4=12,a_{n+1}a_{n-1}=a_n^2\pm 1 $

Show that there is a unique sequence of positive integers $(a_n)$ satisfying the following conditions. $$a_1=1,a_2=2,a_4=12,a_{n+1}a_{n-1}=a_n^2\pm 1$$ I approached the problem to find out, ...
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1answer
57 views

Prove that : $\lvert s_n - \frac \pi 4\rvert \le \frac 1 {2n+1}$, where $s_n = \sum^{n-1}_{j=0} \frac {(-1)^j} {2j+1}$

Prove (Leibniz' series): $|s_n - \frac \pi 4| \le \frac 1 {2n+1}, \forall n \in \mathbb N$ where $s_n = \sum^{n-1}_{j=0} \frac {(-1)^j} {2j+1} = 1 - \frac 1 3 + \frac 1 5$ ... To prove the result ...
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0answers
93 views

How find this sequence such this three condition (AMM Problem 2012)

Prove that for all $n\ge 4$, there exsit integer $x_{1},x_{2},\cdots,x_{n}$ such following conditions, (1):$$x_{1}=1,x_{k-1}<x_{k}<3x_{k-1},2\le k\le n-2$$ (2): ...
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2answers
55 views

Does the Leibniz Test say that if the sequence doesn't go to 0 that the series diverges?

My book explains the Leibniz test by saying: "Assume a sub n is a positive sequence that converges to 0..." And goes on to say that that means the alternating series converges. What if the sequence ...
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2answers
74 views

Why does $\lim_{n \to\infty}a_{n+1} = \lim_{n\to\infty}a_n$?

Assume that $\{a_n\}$ is a convergent sequence. How to use the definition of a limit of a sequence to prove that
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1answer
106 views

Assuming ${a_n}$ is a convergent sequence, prove that the lim inf of $a_{n+1}$ is equal to the lim inf of $a_n$

I'm aware that you have to use the definition of a limit of a sequence, which is: $\lim\limits_{n \to \infty} a_n = L $ if for every $E > 0$, there is an $N$ such that if $n >N $then $|a_n - L ...
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1answer
235 views

Waring's Inequality Solution

$$ \text{Waring's problem asks, "Is }\left\lfloor \left(\frac{3}{2}\right)^n\right\rfloor =\left\lfloor \frac{3^n-1}{2^n-1}\right\rfloor\text{ always true?"} $$ We craft an inequality, with $m,n ...
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1answer
33 views

Is this complex function harmonic?

Let us consider the following convergente series in the set $0<x<1$ and all real $y$: $$h(x+iy)=∑_{n=2}^{∞}(-1)ⁿ⁻¹((n^{2x-1}-1)/n^{x})n^{iy}$$ My question is: Is this complex function ...
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1answer
72 views

Serie of functions : interchange of limit of series

Let $\{f_n\}_{n=1}^\infty$ be a sequence of real-valued functions on $\mathbb{R}$. Show that if $f_n$ is continuous for all $n \in \mathbb{N}$ and the series $\sum_{n=1}^\infty f_n$ converges ...
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3answers
89 views

Determine whether the given series is convergent.

I have a series: $$ \sum_{n=1}^\infty \frac{3^n}{n!}$$ The task is to investigate if this series converges or diverges. I know that if $\lim_{n\to\infty}\frac{3^n}{n!}$ is infinity or a non-real ...
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1answer
45 views

Compact convergence of series

I am trying to show that $$f(z) = \sum_{k=1}^{\infty} \frac{(-1)^{k}}{z+k}$$ converges compactly over $\mathbb{C}$ and starting to think that this statement is false after several attempts. If I ...
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1answer
72 views

Can $ \lim_{n \to \infty} \sum_{i=1}^{n} (1-\frac {x_i}{\sqrt{x_i^2+r^2}}) \cdot (x_{i+1}-x_{i})$ be written as a definite integral

$$ \lim_{n \to \infty} \sum_{i=1}^{n} \left (1-\frac {x_i}{\sqrt{x_i^2+r^2}}\right) \cdot (x_{i+1}-x_{i})$$ $x_1=1$, $x_n=a+L$ I don't really see a way to manipulate this into a desirable form, but ...
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1answer
73 views

Proving the convergence and divergence of the p-series

We know from calculus that $\sum_{n=1}^{\infty} \frac{1}{n^p}$ diverges if $p \in [0,1)$ and converges if $p > 1$. I want to use analysis to prove these two statements. For the case where $p > ...
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0answers
39 views

if $F_{1}=F_{2}=1,F_{n+2}=F_{n+1}+F_{n}$ ,then there exist $F_{i}$,such $p|F_{i}$ [duplicate]

Let sequence $\{F_{n}\}$ such $$F_{1}=F_{2}=1,F_{n+2}=F_{n+1}+F_{n}$$ let $p$ is prime number,show that:there is exsit $F_{i}$ such $$p|F_{i},1\le i\le p+1$$ my idea: if $$p=2$$,then then ...
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2answers
139 views

Is the sequence of function $\frac{\sin nx}{\pi x}$ convergent to $\delta(x)$

I have no doubt that the integral of the function from negative infinity to positive infinity is 1. But I don't think $f_n(x) = \frac{\sin nx}{\pi x} \rightarrow 0 $ as $ n \rightarrow \infty $ given ...
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2answers
91 views

Why can't a Series Converge to 1?

The Divergence Test states: If $\sum a_k$ converges, then $\displaystyle\lim_{k\to\infty}a_k = 0$. Equivalently: If $\sum a_k$ diverges, then $\displaystyle\lim_{k\to\infty} a_k \neq 0$. ...
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3answers
189 views

Continuity of Fixed Point

For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that $\left\| f_a(x)-f_a(y) \right\| \leq ...
2
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0answers
136 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 ...
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4answers
150 views

Find the Exact sum

Give the fourier series representation of $f(x) = x$ on $[-\pi, \pi]$. Use the result to give the exact sum of... $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}$$ $$\text{ where } x \in [-\pi,\pi]$$
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3answers
70 views

Question regarding a sequence and a monotone subsequence

Let $x_n$ be a sequence and $x_0$ a number. How can I show that $x_n$ --> $x_0$ if and only if ${x_n}_k$ --> $x_0$ for every monotone subsequence ${x_n}_k$? Any hints or solutions welcome.
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1answer
42 views

Series Sum up to N terms

I have been trying to find the sum of a series given by $ t(n) = \frac{1}{2^n-1}$, up to N terms. All I could do is to see that the difference of the successive denominators form a GP. Kindly help me ...
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0answers
48 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 ...
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0answers
192 views

A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. Looking at the series $$ s(1/2,2) = (1/2)+(1/2)^4+(1/2)^9+(1/2)^{16}+... $$ and more general $$ s(b,p) ...
1
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1answer
60 views

To prove that $\limsup_{n\to\infty}\frac{1}{a_n} = \frac 1{\liminf_{n\to\infty}a_n}$

I want to prove this equality: $$\limsup_{n\to\infty}\frac{1}{a_n} = \frac{1}{\liminf_{n\to\infty}a_n},\; \forall n\ a_n > 0.$$ What I was thinking about was: Just following the definition, let ...
1
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1answer
52 views

About definition of recursive sequence

Can I define the recursive sequence in the following? Let $f: \Bbb{N} \to \Bbb{R}$ a sequence and $k \in \Bbb{N}$ and $\forall x \in \Bbb{N}(f_x:=f(x))$ , $f$ is recursive sequence if ...
5
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2answers
64 views

How to find the limit of $\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$

How to find the following limit: $$\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$$ where $H(n) = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n^{th}$ harmonic number and $\gamma$ is the Euler ...
3
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1answer
69 views

Problem 6.b, Section 3.2, Introduction to Real Analysis, Bartle

Can somebody explain to me how to compute this limit? $$\lim_{n\to \infty}\frac{(-1)^n}{n + 2}$$
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3answers
43 views

sequence of positive numbers converging to a positive limit

Suppose that {s$_n$} is a sequence of positive numbers converging to a positive limit. Show that there is a positive number c so that s$_n$ > c for all n.
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0answers
32 views

Is there a way to expand Re(Li(a^z)) in series?

I'm searching a way to expand $ f(z) = Re(Li(a^z)), a \in R, z \in C $ in series. The computer-friendly, quickly convergent series is a huge plus. For being 'computer-friendly' I mean a relatively ...