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

learn more… | top users | synonyms (5)

0
votes
3answers
21 views

About the infinite geometrical sequence factored with n

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 ...
0
votes
0answers
13 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 ...
1
vote
0answers
18 views

Infinite series bound

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
1
vote
2answers
19 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" ...
0
votes
2answers
23 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 ...
0
votes
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
votes
1answer
30 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 ...
0
votes
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 ...
5
votes
2answers
76 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 ...
1
vote
1answer
36 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
votes
2answers
69 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 ...
1
vote
0answers
16 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
votes
1answer
22 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?
3
votes
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 ...
2
votes
1answer
20 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 ...
1
vote
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. ...
0
votes
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 ...
0
votes
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
votes
1answer
49 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, ...
1
vote
1answer
64 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!
0
votes
0answers
25 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 ...
-1
votes
0answers
18 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.
2
votes
0answers
71 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 ...
1
vote
1answer
23 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
votes
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
votes
1answer
31 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
votes
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 ...
0
votes
1answer
33 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} ...
0
votes
1answer
29 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. ...
0
votes
1answer
34 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 ...
1
vote
1answer
29 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!
1
vote
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 ...
2
votes
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 ...
1
vote
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 ...
1
vote
3answers
67 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.
0
votes
1answer
23 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: ...
1
vote
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 $\lim_{n\to \infty}(x_n,y_n)$ . ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
11 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
vote
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
votes
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
votes
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 ...
11
votes
3answers
87 views

An equivalent for $\sum_{n=0}^{\infty} e^{-x\sqrt{n}}$ as $x$ tends to $0^+$

I would like to obtain an equivalent form for $$ f(x)=\sum_{n=0}^{\infty} e^{-x\sqrt{n}} $$ as $x \rightarrow 0^+$. I tried without success to "remove" the $\sqrt{\cdot}$ in the summand by summing ...
2
votes
2answers
92 views

Arithmetic progressions.

"Consider an 4 term arithmetic sequence. The difference is 4, and the product of all four terms is 585. Write the progression". My way of finding the progression seems like it will take too long, but ...
0
votes
1answer
31 views

Voltage Distribution Inside a Cylinder [on hold]

I was assigned this problem, and quite honestly I do not know where to begin. If I could get some help and an explanation of the Bessel function, also? Thank you. I know my conditions are: ...
0
votes
0answers
17 views

Prove series converge using comparison test

I got a question which is Suppose that $a_n \ge 0 $ $\forall n \in \mathbb{N}$ and that $\sum^{\infty}_{n=1}a_n$ converges. Prove that $\sum^{\infty}_{n=1}(a_n)^2$ also converges. And what I did is ...
1
vote
1answer
29 views

prove that $n^\epsilon$/log(n) goes to infinity without derivatives or functions

I'm looking for a way to prove that for every $\displaystyle{\quad\epsilon\ >\ 0\,,\quad{n^{\epsilon} \over \log\left(\, n\,\right)} \to \infty,\quad}$ treating it only as a sequence, without using ...
0
votes
2answers
26 views

A sequence $(a_n)$ where $\exists M>0$ such that $\forall n\in\mathbb{N}$, $\sum\limits_{k=1} ^n |a_{k+1}-a_k|\leq M$. Show $(a_n)$ is Cauchy sequence

Question: Let $(a_n)_{n\in\mathbb{N}}$ be a sequence where there exists $M>0$ such that for all $n\in\mathbb{N}$, $\sum\limits_{k=1} ^n |a_{k+1}-a_k|\leq M$. Show that $(a_n)_{n\in\mathbb{N}}$ ...