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

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0
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1answer
59 views

Convergence of the sum of the subsequence $\sum_{k=1}^{\infty } 2^{k}x_{n_{k}}$

Let $\{x_{n}\}$ be a sequence such that $\lim_{n \to ∞}x_{n}=0$. Prove that there exists a sub sequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $\sum_{k=1}^{\infty } 2^{k}x_{n_{k}}$ converges and ...
3
votes
1answer
52 views

Estimating a series $\sum_{k=1}^\infty e^{-k^2t}\leq \frac{1}{2}\sqrt{\frac{\pi}{t}}?$

Can we prove such an estimate $$\sum_{k=1}^\infty e^{-k^2t}\leq \frac{1}{2}\sqrt{\frac{\pi}{t}}?$$ I need it in my research....
8
votes
2answers
255 views

How find this arithmetic sequence of $n$

if there exist positive integer sequence $a_{1},a_{2},a_{3},\cdots,a_{n}$,such that $$a_{1}a_{2},a_{2}a_{3},a_{3}a_{4},\cdots,a_{n-1}a_{n},a_{n}a_{1}$$ is arithmetic sequence,and the common ...
2
votes
1answer
49 views

Convergence of $\frac{1}{1^b}+\frac{1}{2^b}+\frac{a}{3^b}+\cdots+\frac{1}{(3k-2)^b}\frac{1}{(3k-1)^b}\frac{a}{(3k)^b}+\cdots$

For which values of $a,b\in\mathbb{R}$ the series: $$\frac{1}{1^b}+\frac{1}{2^b}+\frac{a}{3^b}+\cdots+\frac{1}{(3k-2)^b}\frac{1}{(3k-1)^b}\frac{a}{(3k)^b}+\cdots$$ converges? Attempt: ...
0
votes
2answers
233 views

How should I handle the limit with the floor function?

I wanted to show that $$ \lim_{n\to\infty} \bigg( \frac{n+\lfloor x \sqrt{n} \rfloor}{n-\lfloor x \sqrt{n} \rfloor}\bigg)^{\lfloor x \sqrt{n} \rfloor} =e^{2x^2} $$ After applying $x \sqrt{n} -1 \leq ...
3
votes
2answers
139 views

Convergence of $\sum_{n=1}^{\infty} \frac{\ln n }{\sqrt{n^3-n+1}}$

Why the following series $$\sum_{n=1}^{\infty} \frac{\ln n }{\sqrt{n^3-n+1}}$$ converges?
1
vote
1answer
37 views

What does $\sum_{n\in\mathbb N}\frac{n^x}{x^n}$ converge to when $x\in\mathbb R^+$ and $x>1$?

What does $\sum_{n\in\mathbb N}\frac{n^x}{x^n}$ converge to when $x\in\mathbb R^+$ and $x>1$? I'm looking for a hint of how to tackle this problem.
1
vote
2answers
555 views

calculate riemann sum of sin to proof limit proposition

$$\lim_{n \to \infty}\frac1n\sum_{k=1}^n\sin(\frac{k\pi}{n})$$ I'm having trouble expressing $\sin(x)$ differently here in order to calculate the riemann sum. I want to show that this converges to ...
26
votes
5answers
3k views

Find closed form for $1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, \ldots$

Is there any closed form for the following? $$1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, \ldots$$ I tried to find one, but I failed. I saw solution on Wolfram Alpha, but I didn't understand it: ...
3
votes
3answers
298 views

$\sum_{n=1}^{\infty }\left(\frac{2n+5}{7n+6}\right)^{n\log(n+1)} $ converges or diverges?

I am trying to determine whether this series converges or diverges: $\sum_{n=1}^{\infty }\left(\frac{2n+5}{7n+6}\right)^{n\log(n+1)}$. Here is my solution: I called: ...
9
votes
3answers
328 views

Integral of polylogarithms and logs in closed form: $\int_0^1 \frac{du}{u}\text{Li}_2(u)^2(\log u)^2$

Is it possible to evaluate this integral in closed form? $$ \int_0^1 \frac{du}{u}\text{Li}_2(u)^2\log u \stackrel{?}{=} -\frac{\zeta(6)}{3}.$$ I found the possible closed form using an integer ...
4
votes
1answer
159 views

hard time with series convergence or divergence [duplicate]

I'm having real hard time with this series I can't prove that the series converges and also I can't prove that the series diverges: $$\sum_{k=1}^\infty\frac{\sin^2(n)}{n}.$$ any help would be ...
4
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2answers
84 views

Alternative unconditional form of $\sqrt{n -\sqrt{n -\sqrt{n -\cdots}}}$?

Consider $a_n$, where $$\begin{align} a_n &=\small{\sqrt{n -\!\!\!\sqrt{n -\!\!\!\sqrt{n -\!\!\sqrt{n -\!\!\sqrt{n -\!\!\sqrt{n -\!\sqrt{n - \cdots}}}}}}}}\end{align}$$ Using a recursive ...
0
votes
3answers
31 views

arithmectic sequence

what is the 100th trerm of the arithmectic sequence? 3,8,13,18 can you help me find it, i know there a formula but i dont rember what it is !!!!!! ive know you have to pluge it in to some kind of ...
2
votes
1answer
56 views

Limit of sequence $u_1,u_3,u_5,\dots$ with $u_{n+1}=1+\frac{1}{u_n}$

We have a sequence of numbers defined recursively by $$u_{n+1}=1+\frac{1}{u_n},$$for $n\geqslant 1$. It is also given that $u_1=1$. Find the limit $l$ of the sequence $u_1,u_3,u_5,\dots$. So I ...
-2
votes
1answer
107 views

Is this just a version of the binomial theorem?

I asked a question related to it and found something interesting (at least that is what I think)... Here is the link to the original question: What is the pattern of this sequence? I went through a ...
12
votes
4answers
624 views

Does $\lim \frac {a_n}{b_n}=1$ imply $\lim \frac {f(a_n)}{f(b_n)}=1$?

I wanted to prove the seemingly simple statement: If $\lim \frac {a_n}{b_n}=1$ and $f$ continuous with $f(b_n)\neq0$ then $\lim \frac {f(a_n)}{f(b_n)}=1.$ I started promptly with \begin{align} ...
1
vote
2answers
457 views

Difference between divergent series and series with no limit?

Can a series have a limit and be divergent? I'm confused about the difference between divergence and a series not having a limit. A ck-12 calculus book stated they are different concepts.
8
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2answers
532 views
2
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1answer
1k views

A bounded sequence in $L^\infty$ has a weak-$^*$ convergent subsequence

Suppose $u_n$ is bounded in $L^\infty(\Omega)$. $\|u_n\|_{L^\infty(\Omega)}<M$. Then $u_{n_k}\to u$ weak star in $L^\infty(\Omega)$ for some $n_k\uparrow \infty$ and $u\in L^\infty$. I want to ...
1
vote
1answer
118 views

Convergence of Monotone Sequence in Affine-Extended Reals

For a monotonically increasing sequence $(x_n)$ in $\mathbb{R}$, we have that $\lim\limits_{n \to \infty} x_n$ exists and is finite iff $(x_n)$ is bounded, i.e. $\lim\limits_{n \to \infty} x_n = \sup ...
0
votes
1answer
106 views

Joining finite sequences [duplicate]

How do I describe the joining of two finite sequences in mathematical notation? For example, suppose the following: $$ A=(a_i)_{i=1,2}=(4,2)\\ B=(b_i)_{i=1,2}=(9,5)\\ C=(c_i)_{i=1,...,4}=(4,2,9,5) $$ ...
0
votes
1answer
51 views

Is the sum of the reciprocal of full reptend primes convergent?

Given: The sum of the reciprocals of the primes is divergent. Given: The sum of the reciprocals of the twin primes is convergent.
0
votes
3answers
78 views

Summing a series - Calculus 1.

I'm learning Calculus 1 at the collage, and the semester's end is close, which bring with it the exams period. So I pretty much understand all the topics, except for a series summing. I don't know ...
2
votes
2answers
42 views

adding infinitely many equations side by side in a recurrence relation

we are given that $x+\beta y_{n+1}=k_n+y_n$ for all $n\in\mathbb{N}\cup\{0\}$, where $\beta\in(0,1)$, $y_0=0$, and $k_n$ is 6 whenever $n$ is even and 4 whenever odd. Being the naive mathematician I ...
7
votes
1answer
94 views

Why does $\sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )+{1\over2}-{1\over3} = \gamma$?

How could one prove that $$x = \sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )$$ is such that $x+{1\over2}-{1\over3} = \gamma$ ? I am having problems just calculating ...
2
votes
1answer
41 views

Analysis of $\sum_{k=1}^{\infty}\frac{x^2}{(1+x^2)^k}$

Given the sum $\sum_{k=1}^{\infty}\frac{x^2}{(1+x^2)^k}$, I have two questions. (Sorry for that, it's for my exam preparations): Prove that $\sum_{k=1}^{\infty}\frac{x^2}{(1+x^2)^k}$ converges ...
1
vote
3answers
547 views

How to show convergence or divergence of a series when the ratio test is inconclusive?

Use the ratio or the root test to show convergence or divergence of the following series. If inconclusive, use another test: $$\sum_{n=1}^{\infty}\frac{n!}{n^{n}}$$ So my first instinct was ...
1
vote
1answer
57 views

Is a bounded sequence Cauchy if the element come closer?

MISSED THE CONDITION ON THE SUP.... I try to prove the existence of a limit in a Banach space. I have a sequence $\{x_n\}$ and I have managed to prove that $\limsup_{n\to\infty}\|x_n\|= C<\infty$ ...
0
votes
1answer
93 views

When is an infinite sequence of integers purely deterministic with no randomness involved?

I see in literature very different descriptions of what is a deterministic system such as: "... a system in which no randomness is involved in the development of future states of the system...>>>" I ...
0
votes
1answer
78 views

What is the pattern of this sequence?

I went though this pattern and I think the results might be interesting. It was a long one but I'm only showing the first five (to make things look simpler). $$0,1,a+b,a^2 + b^2 + \frac 32ab , ...
0
votes
2answers
136 views

I don't understand this notation… - Series with ln

I found this notation in my book $$ \sum\limits_{i=1}^n \ln^n3 $$ and I don't know how to interpret it. Is it $$ \sum\limits_{i=1}^n \ln((1^n)\cdot3)\;? $$ And btw, how to check if this series ...
4
votes
3answers
149 views

Determine if $\sum^{\infty}_{n=1}\int^{\frac{\sin n}{n}}_0\frac{\sin x}{x} \, dx $ is converges or diverges.

Determine if $\sum^{\infty}_{n=1}\int^{\frac{\sin n}{n}}_0\frac{\sin x}{x} \, dx $ is converges or diverges. Could someone please show me how to do it? And especialy showing why ...
2
votes
2answers
88 views

Need help applying the root test

I'm not sure if I am doing something wrong, or not... I've got an answer but it doesn't look right to me. Given the following series, determine if it is convergent or divergent using the root or ...
2
votes
1answer
109 views

Is this generalization of Dirichlet's test true?

Dirichlet's Test for Series is stated as follows: Let $(a_n)_{n=1}^{\infty}$ be a monotone decreasing or increasing sequence that converges to $0$. Let $(b_n)_{n=1}^{\infty}$ be another sequence ...
1
vote
1answer
63 views

Develop five terms in the Taylor series

Develop five terms in the Taylor series around $x_0=\pi$ for the function $f(x)=\cos\left({x\over3}\right)$ $f^0(x)=\cos\left({x\over3}\right) \Big|_\pi $ $f^{'}(x)=-\sin\left({x\over3}\right) ...
3
votes
1answer
112 views

Why doesn't this sequence have any pointwise subsequence?

Prove that the sequence of functions $f_n(x) = \sin(n x)$ have no pointwise convergent subsequence I am confused. If I let $x = 1$, then we get $f_n(1) = \sin(n)$. By Bolzano Weistress, we have a ...
0
votes
1answer
128 views

Closed form solution for $a_n = 1,11,111,1111,…$

I'm trying find a closed form for the sequence $$\begin{matrix} a_1&1\\ a_2&11\\ a_3&111\\ \cdots&\cdots\\ a_9&111111111\\ a_{10}&12222222121\\ \cdots&\cdots ...
3
votes
5answers
175 views

Testing convergence of $\sum_{n=0}^{\infty }(-1)^n\ \frac{4^{n}(n!)^{2}}{(2n)!}$

Can anyone help me to prove whether this series is convergent or divergent: $$\sum_{n=0}^{\infty }(-1)^n\ \frac{4^{n}(n!)^{2}}{(2n)!}$$ I tried using the ratio test, but the limit of the ratio in ...
1
vote
2answers
98 views

Is $\sum_{k=4}^{\infty }{k^{\log(k)}}/{(\log(k))^{k}}$ convergent or divergent?

I came across this problem in a textbook, and the question is to investigate the convergence/divergence of the following series: $$\sum_{k=4}^{\infty }\frac{k^{\log(k)}}{(\log(k))^{k}}$$. I have no ...
2
votes
2answers
94 views

About the convergence of $\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{2}+n^{1/\gamma}}$

Does the series converge? $$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{2}+n^{1/\gamma}}$$
3
votes
2answers
102 views

A problem on continuity of a function on irrationals for $f(x) = \sum_{r_n \leq x} 1/n^2$ [duplicate]

Let $\langle r_n\rangle$ be an enumeration of the set $\mathbb Q$ of rational numbers such that $r_n \neq r_m\,$ if $\,n\neq m.$ $$\text{Define}\; f: \mathbb R \to \mathbb ...
1
vote
1answer
73 views

Series summation proof

I deduced that $$\frac{1}{e^x-1}\sum_{s=1}^{n-1}\frac{1}{e^x-e^{\frac{2\pi i s}{n}}}=\frac{\sum_{s=1}^{n-1}(n-s)e^{(n-s-1)x}}{e^{nx}-1}$$ Now, I want to get an analytic proof of this deduced result.
0
votes
1answer
62 views

Upper bound to a series with ceiling

If we know that: $$a_{i+1} \leq {a_i \over 2}$$ Then we can calculate an upper bound for every n: $$a_{n} \leq {a_0 \over 2^n}$$ But what if we keep the elements integer, by taking the ceiling: ...
2
votes
1answer
83 views

How to calculate a bound for this product?

Consider the following product: $$ \prod_{i=1..n} {\left(1 - {1 \over 2^i}\right)} $$ A numeric calculation, up to $n=20$, gives $0.288788370496567$. But how can I calculate its limit when $n$ goes ...
8
votes
2answers
543 views

Proving a function is continuous on all irrational numbers

Let $\langle r_n\rangle$ be an enumeration of the set $\mathbb Q$ of rational numbers such that $r_n \neq r_m\,$ if $\,n\neq m.$ $$\text{Define}\; f: \mathbb R \to \mathbb ...
1
vote
2answers
54 views

A problem on sequences and series

Let $ (x_n) $ be a sequence of real numbers such that $ \lim x_n =0 $. Prove that there exists a subsequence $(x_{n_k} )$of $ (x_n) $ such that $ \sum_{k=0}^\infty 2^{k}x_{n_k}$ coverges and ...
1
vote
3answers
56 views

Boundedness of $\sum_{m=k}^{\infty} \frac{k}{m^2}$

Is the series $\sum_{m=k}^{\infty} \frac{k}{m^2}$ bounded independently of k?
2
votes
0answers
51 views

Least upper bound for a positive real sequence satisfying $|x_u−x_v|\cdot|u−v|>1$

The starting point of this question is the: IMO 1991, problem 6: Prove that for any $\alpha>1$, there exist a bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ such that, if $u,v$ are distinct ...
0
votes
4answers
126 views

Why does $\frac{1}{n+\ln n} ∼ \frac{1}{n}$?

I have to find if the sequence $\frac{1}{n+\ln n}$ is convergent or divergent? And in the correction they've just wrote : $\frac{1}{n+\ln n} ∼ \frac{1}{n}$ ??