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

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1answer
12 views

Show that a Series Diverges

Question: Let a sequence ($a_n$) have the property $\lim \limits_{n \to \infty} na_n = a > 0$ Show that the series $\sum_{n=1}^\infty a_n$ diverges Attempts: Basically, I firstly tried ...
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5answers
37 views

Prove Using Definition that the Sequence $\frac{n+1}{\sqrt{n}+2}$ Diverges to Infinity

Formally, a sequence $x_n$ diverges to infinity whenever for all $M>0$ there exists $N(M)$ such that $n>N(M)$ implies $x_n>M$. Prove formally, using the definition, that the following ...
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1answer
32 views

Prove $\cos x = \frac{8}{\pi}\sum_n \frac{n\sin 2nx}{4n^2-1}$ with Fourier series

I want to prove $$\cos x = \frac{8}{\pi}\sum_n \frac{n\sin 2nx}{4n^2-1}\;x\in(0,2\pi)\;\;\;\;[1]$$ I have two questions regarding this: $(1)$ How can I find a function $f$ such that the former ...
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1answer
42 views

Proof that it is not uniformly convergent on R [on hold]

Prove that the series $$\sum_{n=1}^\infty 2^n \sin \left( \frac{x}{3^n} \right)$$ is not uniformly convergent on R.
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1answer
78 views

Evaluate $\sum\limits_{n=1}^{\infty} \frac{2^n}{1+2^{2^n}}$

How to evaluate the infinite series: $$\sum\limits_{n=1}^{\infty} \frac{2^n}{1+2^{2^n}}$$
3
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2answers
73 views

$\frac{1}{x^2} \int xe^x dx$ without using integration by parts

On a test i just had, i needed to solve a differential equation which lead me to having to find the result of $$ \frac{1}{x^2}\int xe^x dx $$ I then attempted to do this integral without integration ...
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3answers
39 views

About the infinite geometrical sequence factored with n [duplicate]

I just came across this thread, and i asked myself: I know that $\sum^\infty_{n=0} x^n = \frac{1}{1-x}$ But what happens when we set up the sum like $$\sum^\infty_{n=0} nx^n = ?$$ There is ...
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0answers
20 views

Series in closed form

Let $\alpha \in (0,1)$ and $z\geq 0$. Define the function: $$ \theta_{\alpha}(z)=\sum_{j=0}^{\infty}\left(\frac{z}{\alpha j + 1}\right)^j. $$ Can $\theta_{\alpha}(z)$ be written in closed form? Are ...
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1answer
35 views

The asymptotic behavior of $\sum_{n=1}^\infty\frac{1-\cos(x4^n)}{2^n}$ as $x\to 0$

Is there a way to show that for small $x$'s $$\sum_{n=1}^\infty\frac{1-\cos(x4^n)}{2^n}\le c\sqrt x$$ I tried Taylor expansion of $\cos$ and square root... Thank's
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1answer
25 views

How to prove matrix geometric convergence to any matrix?

Suppose I have two vectors $x$ and $v$, and we want to calculate the following expression: $$(I+x\cdot v^{T})^{-1}$$ My professor affirmed that we could treat this as a "geometric progression" ...
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2answers
24 views

How to check convergence of the following series

How to check convergence of: $1.\sum_{n=1}^\infty \frac{(n+1)^n}{n^{n+\frac{5}{4}}}$.I tried using Cauchy's root test but got limit=1.How to do it? $2.\sum_{n=1}^\infty \frac{1}{n^{1/2}}tan ...
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0answers
17 views

When does the limit of the ratio of consecutive terms of a sequence exist?

I am trying to understand and obtain some sufficient conditions under which the limit of the ratio of consecutive terms of a sequence exists. Let $x_n$ be a sequence of positive integers, such that ...
0
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1answer
32 views

Convergent squence in topology

Please, I consider this topological space $(E,\tau)$ where $\tau=\{G\subset E, ~\text{card}~ (E\setminus G)~\text{countable}\}\cup\{\emptyset\}$ How to prove that a sequence $(x_n)$ is convergent in ...
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1answer
16 views

How is OEIS sequence A120933 'maximal leading nondecreasing subword ' to be understood?

For n=2 we only have these four binary words: 00 01 10 11 What is the procedure for calculating by hand T(2,1) and T(2,2)? I'm trying to understand the reasoning behind this sequence as I can't see ...
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2answers
81 views

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

During my calculation I ended with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express in term of series by taking the logarithm ...
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1answer
37 views

Infinite series and the Riemann zeta function

I have two questions concerning infinite series in the context of the Riemann zeta function. Given the properties of infinite series, why can't we regroup the terms in $\zeta(0)$ in such a way as to ...
4
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2answers
71 views

prove the limit of $k^{1/k}$ is $1$ [duplicate]

I want to prove that the limit of the sequence $k^{1/k}$ is $1$ as $k$ tends to infinity without using advanced rules such as L'Hospital's Rule and just using the basic rules in real analysis. How ...
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0answers
22 views

finding a function whose values at certain points are defined

I have certain polynomials in a sequence from which I am trying to derive the common term. The polynomials are: $7k^2-8\\ 21k^4-368k^2-704\\ ...
0
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1answer
23 views

Sequence word problem [on hold]

A basketball is dropped from 81 meters atop the tower. If it rebounds up 2/3 of the distance after each bounce, what is the total vertical distance traveled by the ball before it come to rest?
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1answer
21 views

Convergence of monotone $f_n:[0, \infty) \rightarrow [0,1]$ to continuous, monotone $g$ is uniform

Let $g: [0, \infty) \rightarrow [0,1]$ be a continuous, monotone increasing function where $g(0)=0$ and $g(x)\rightarrow 1$ as $x \rightarrow \infty$. Also let $f_n:[0, \infty) \rightarrow [0,1]$ be a ...
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1answer
22 views

For $s_n$ a sequence in $\Bbb R$, if $\lim s_n$ defined as a real number, then $\liminf s_n = \lim s_n = \limsup s_n$.

I'm studying some real analysis, and I'm trying to figure out how to prove the following theorem. For the most part, I've got things figured out. I'll post the theorem and my proof (which closely ...
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2answers
147 views

Finding Sum for Infinite Series

Normally when I keep try multiple ways to solve a problem, I get an idea of where to start, and eventually can solve it. But it hasn't been the way for this question and I've been stuck for hours. ...
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0answers
11 views

Tips on effectively representing this recurrence relation in a generalized form.

The recurrence relation is $$ y_n = d_1 y_{n-1}+\frac{d}{dx}[y_{n-1}] $$ a good thing to note as well is $$ \frac{d}{dx}[d_n] = d_{n+1} $$ This is terrible to expand out after a good while, The main ...
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3answers
31 views

Prove that $s_n\leq 2-\frac{1}{n}$ for all $n\geq 1$

We have the sequence $(s_n)_{n\geq 1}$ given by $s_n=\sum^n_{k=1}\frac{1}{k^2}$. Prove that $s_n\leq 2-\frac{1}{n}$ for all $n\geq 1$. Thanks in advance!
5
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1answer
51 views

Proof that the harmonic series is < $\infty$ for a special set..

In one of my books i found a very interesting task, i am really curios about the solution: Let $M = \{2,3,4,5,6,7,8,9,20,22,...\} \subseteq \mathbb{N}$ be a set that contains all natural numbers, ...
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1answer
65 views

Give an example where $\{a^2_n\}_{n=1}^\infty$ is a Cauchy sequence, but $\{a_n\}_{n=1}^\infty$is not.

Give an example where $\{a^2_n\}_{n=1}^\infty$ is a Cauchy sequence, but $\{a_n\}_{n=1}^\infty$is not. Could somebody give me an example of this? Thanks in advance!
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0answers
26 views

Convergence of $|f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)}$ on $x \in [0,1]$

I have the following relationship between $f_n$ and $f$ on $x \in [0,1]$ \begin{align*} |f_n(x+n)-f(x+n)| \le \frac{\log\log \left(1+\frac{x+n}{\gamma}\right)}{\log(1+n)} \end{align*} for all $x \in ...
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0answers
19 views

Series comparison test for $ \sum_1^\infty n^{\ln(n)} \ln(n^n)$

$$ \sum_{n=1}^\infty n^{\ln(n)} \ln(n^n)$$ Which function should I use to compare this to proove that it diverges? To me comparison test for this series the obvious solution.
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0answers
77 views

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$, where $\left(\binom{a}{b}\right)=\dfrac{a!!}{b!!(a-b)!!}$ EDIT : Someone pointed out in the Mathematics chat that my ...
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1answer
25 views

Product of a Convergent Series and Bounded Sequence

Let $a_n$ be a bounded sequence and $\sum_{n=1}^\infty b_n$ be a convergent series. Then $\sum_{n=1}^\infty b_na_n$ is convergent. I have found a counterexample to prove it false; If we let ...
2
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2answers
35 views

First five terms of the sequence $a_n=2/e^n$ [on hold]

I just wanted to check if my answer here is correct. I am not sure if I am supposed to simplify (e) even further down, and turn the fraction into a decimal or not because it does mention use ...
2
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1answer
33 views

Convergence of the integral of cos(x)/x^2

For $n$ in the natural numbers let $a_n = \int_{1}^{n} \frac{\cos(x)}{x^2} dx$ Prove, for $m ≥ n ≥ 1$ that $|a_m - a_n| ≤ \frac{1}{n}$ and deduce $a_n$ converges. I am totally stuck on how to even ...
2
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1answer
16 views

Prove that if $\sum g_k$ converges uniformly on a set $S$ and if $h$ is a bounded function on $S$, then $\sum hg_k$ converges uniformly on $S$

Prove that if $\sum g_k$ converges uniformly on a set $S$ and if $h$ is a bounded function on $S$, then $\sum hg_k$ converges uniformly on $S$ A little confused about this question, would love to ...
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1answer
34 views

Finding limits superior and inferior of a sequence

Find the limits superior and inferior of the following sequences: note: "For a set, those are the infimum and supremum of the set's limit points, respectively. " $a_n=\frac{n}{n+1} ...
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1answer
31 views

Weak Convergence and its Relationship to a Sequence of Norms

"Prove that $x_n\xrightarrow{w} x$ implies lim inf$||x_n|| \geq ||x||$." I'm trying to understand weak convergence better through this exercise. Here, $\xrightarrow{w}$ means weakly convergent, i.e. ...
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1answer
35 views

Functions or sequence divergences link

One can prove that the sequence $ u_n=\{\sin(n)\}_{n \in \mathbb{N}} $ diverges using a similar argument as in : Proves the divergence of sequence of sin(n) But we can also prove that ...
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1answer
31 views

A property of a sequence

Why if a sequence $a_{n}\rightarrow + \infty$, then it can't be that ${\frac{a_{n+1}}{a_{n}}\rightarrow 0 }$? Thanks!
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3answers
127 views

Show sequence is convergent and the limit

Given the sequence $$\left\{a_n \right\}_{n=1}^\infty $$ which is defined by $$a_1=1 \\ a_{n+1}=\sqrt{1+2a_n} \ \ \ \text{for} \ n\geq 1 $$ I have to show that the sequence is convergent and find ...
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1answer
69 views

Limit of $\frac{2^n}{3^{n+1}}$

I am given the sequence $$a_n=\frac{2^n}{3^{n+1}} $$ I have to show the limit for this sequence, and i think i have gotten the point of it, but i am struggling with how to show it. What i did it ...
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2answers
50 views

Divergent to $\infty \Rightarrow$ Divergent?

In our lecture, we defined a sequence $\left(a_n\right)_{n\in\mathbb N}$ to be divergent if it does not converge, and additionally to be divergent to $\pm \infty$, iff: $$\forall \epsilon \in \mathbb ...
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3answers
72 views

$\frac {1}{5}-\frac{1.4}{5.10}+\frac{1.4.7}{5.10.15}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot$ [on hold]

How to find the sum of the following series: $\frac {1}{5}-\frac{1.4}{5.10}+\frac{1.4.7}{5.10.15}-\cdot\cdot\cdot\cdot\cdot\cdot\cdot$ Any hints.
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1answer
26 views

Demonstrate series of Maclaurin

Find the Maclaurin series of $$f(x)=xe^x$$ Integrate this series term by term in the closed interval $[0,1]$ and demonstrate that: $$\sum^\infty_{2} \frac{1}{(n-2){} !n} = 1$$ I tried it: ...
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0answers
38 views

Finding the convergent value

The sequence is defined as follows : Start : $(x_0,y_0)$ with $ 0 < x_0 < y_0 $ Step : $x_{n+1} = \frac {x_n+y_n} {2}$ , $y_{n+1}= \sqrt{x_{n+1}y_n} $ Find $\lim_{n\to \infty}(x_n,y_n)$ . ...
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3answers
30 views

Functions of sequences and convergence

(a) If $f$ is continuous on $[0,\infty)$ and {$x_n$} is a sequence in $(0,\infty)$ such that {$f(x_n)$} diverges to $\infty$, then $\lim_{n \to \infty} x_n = \infty$. (b) If $f$ is continuous on ...
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1answer
18 views

Alternating Series Test for Divergence

Alternating Series Test if the alternating series $$\sum^\infty_{n=1}(-1)^{n-1}b_n=b_1-b_2+b_3-b_4+b_5+...\;\;\;\;\;b_n\gt0$$ satisfies $$(\text{i)}\;\;b_{n+1}\leq b_n\;\;\;\;\;\text{for all ...
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2answers
24 views

continuity and sequences

If $f$ is continuous on $[a,b]$ and {${x_n}$} is a sequence in $(a,b)$, then {$f$(${x_n}$)} has a convergent subsequence. True or False? If true, prove. If false, give a counterexample. I'm guessing ...
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0answers
12 views

$p_n(x)=a_nx^2+b_nx+c_n$,$a_n,b_n,c_n\in\mathbb{R}\forall n\ge 1$

$p_n(x)=a_nx^2+b_nx+c_n$ be a sequence of quadratic polynomials, $a_n,b_n,c_n\in\mathbb{R}\forall n\ge 1.\lambda_0,\lambda_1,\lambda_2$ are dstinct reals $\ni$ $\lim p_n(\lambda_0)=A_0,\lim ...
1
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2answers
57 views

Trying to find the closed form for the nth term of $\frac{1}{1-x^4}$

I know that $\frac{1}{1-x^4}$ is the generating function for the sequence (1, 0, 0, 0, 1, 0, 0, 0, 1, ...) I don't know how to find the closed form for the nth term though. Itried messing around with ...
0
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2answers
34 views

Discuss the convergence of the sequence who's nth term is given by…

$$a_n = \left(1-\frac{1}{2n}\right)^n$$ Please explain the process of how this is solve, I'm really confuse and struggling on how to figure out series and sequences. Since this is a sequence, is ...
0
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1answer
39 views

Prove that a series is convergent

I have a series which is as follows $$\sum_{n=1}^{\infty}\left(\left(1+\frac{1}{n^3}\right)^n-1\right)$$ and I am asked whether it converges or diverges. I think this series is convergent and I ...