For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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1answer
48 views

Natural ordering with respect to sequences

I came across this question at the beginning of a real analysis book: Under the natural ordering, which of the following are sequences? (a) all integers (b) all integers ≥− 100 (c) all integers ≤ ...
2
votes
1answer
86 views

Finding lim sup and lim inf of a sequence

Consider the sequence $x_1=\frac13,x_{2n}=\frac13x_{2n-1},x_{2n+1}=\frac13+x_{2n}$ where $n\in \mathbb N.$ Explicitly, this sequence is ...
2
votes
1answer
111 views

determine the number of terms in a fibonacci sequence that are divisible by $3$

Consider the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, . . .$ where each term, after the first two, is the sum of the two previous terms. How many of the first $1000$ terms are divisible by 3?
0
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1answer
44 views

Collatz sequences meet at a point

I was solving the problem,in which we are given two starting values of collatz sequence and our task is to say after how many steps their sequences “meet” for the first time. For ex - a= 7 , b= 8 a ...
25
votes
2answers
373 views

Find the closed form of $\sum_{n=1}^{\infty} \frac{H_{ n}}{2^nn^4}$

One of the possible ways of computing the series is to obtain the generating function, but this might be a tedious, hard work, pretty hard to obtain. What would you propose then? ...
0
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2answers
45 views

If $\{N_k\}_{k=1} \subset \Bbb{N}$ is a strictly increasing sequence of natural numbers then $(N_k) \rightarrow \infty$ as $k \rightarrow \infty$

Need to prove: If $\{N_k\}_{k=1} \subset \Bbb{N}$ is a strictly increasing sequence of natural numbers then $(N_k) \rightarrow \infty$ as $k \rightarrow \infty$ I think I just need to show that a ...
1
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3answers
69 views

Convergence of the sequence $\frac{n(n^{1/n}-1)}{\ln(n)}$

I am trying to find the limit of the following sequence $$\lim_{n\to \infty} \frac{n(n^{1/n}-1)}{\ln n}$$ I have tried to use L'Hospitals rule but that did not work. Any pointers of methods to try ...
1
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1answer
53 views

Series involving a Logarithm

Consider the series \begin{align} \sum_{n=1}^{\infty} \left[ \frac{n}{a} \ln\left(1 + \frac{a}{n}\right) - 1 + \frac{a}{2n} \right]. \end{align} Is there a closed form solution to this series and what ...
6
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2answers
166 views

How to show that $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m))) + \cdots$, converges for every real number $m$?

I'm trying to show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m)) + \cdots$ converges for all real numbers $m$. To be specific, the series is defined as follows: $\sum_1^\infty{a_k}$ ...
0
votes
1answer
46 views

Evaluating the limit of a sequence

I'm working my way through some practice problems (no solutions given) for an upcoming exam, and I came across the following problem: Let $A_n = \{s : 0 < s \le \frac{1}{n}\}$. What is the limit ...
1
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1answer
39 views

Convexity of a finite Sequence

I was wondering how to determine whether the following set is convex or not: All $A_1, A_2,..., A_m $are convex sets: $ \sum^m_{i=1} A_i = \{ x_1+x_2+...+x_m,$ $ x_i \in A_i, i=1,2,...,m\}$ can I ...
0
votes
1answer
34 views

Notation in sequences of random variables

I read in statistics books that a sequence of random variables are often written as ${X_n}$. But in all the theorems it just says $X_n$. Why is that? And does ${X_n}$ symbolize the WHOLE sequence ...
1
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1answer
52 views

Compactness, Convexity, Convex Hull of Sets including sequences

Is the following set compact, is it convex and what is the convex hull? $V = \{(x_1, x_2,...,x_n) \in \mathbb{R}^n :\frac{1}{1 + i} \leq x_i \leq \frac{1}{i}, i=1,2,...,n\}$ My thoughts: I was ...
1
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3answers
107 views

Divergence of $n!^{1/n}$ [duplicate]

I'm trying to find a simple proof that $n!^{1/n}$ is a divergent sequence. I have proved it using a lower bound you can get from an integral (or Stirling's approximation) that $n!^{1/n}>\ln(n)$, ...
11
votes
1answer
205 views

Compute polylog of order $3$ at $\frac{1}{2}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{2^nn^3}$$ I am aware this equals polylog of order $3$ at $\frac{1}{2}$ or $\operatorname{Li}_3\left(\frac{1}{2}\right)$, but how ...
1
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1answer
57 views

Prove that x{n} is convergent

So I'm currently studying for my midterms and I found the following question from my practice set which I'm unable to solve: Prove that the following sequence $(x_{n})$ is convergent. Let $$x_{n} = ...
0
votes
1answer
89 views

Do the first k numbers of geometric progression cover all possible numbers?

Original problem: we are looking for up to first ten digits of decimal representation of powers of 2 $$\require{cancel}2^0 = 1 \to \{1\}\\ 2^1 = 2 \to \{2\}\\ ...\\ 2^{10}=1024\to\{1,10,102,1024\}\\ ...
2
votes
2answers
114 views

Prove that there exists a natural number $K$ such that $a_{n} < b_{n}$ for all $n \geq K$

Given $\displaystyle\lim_{n\rightarrow \infty} a_{n}=a$ and $\displaystyle \lim_{n\rightarrow\infty} b_{n}=b$, and we have $a < b$, how does one prove that there exist a natural number $K$ such ...
0
votes
2answers
341 views

Prove that limit exists and is finite, sum of series, Cauchy.

Prove that $\lim_{n \to \infty} \sum_{k=1}^n \frac{(-1)^k}{k}$ exists and is finite. Attempt: Suppose $\{x_n\}$ is real sequence, and $x_n = \frac{(-1)^k}{k}$. I know if I prove that it is Cauchy, ...
1
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1answer
434 views

limsup of the product of two sequences, of which one converges

Let $\{a_n\}_{n=1 }^{\infty}$ and $\{b_n\}_{n=1}^{\infty} $ be two sequences in $\mathbb{R}$, with the first sequence convergent . Prove that $$ \limsup\limits_{n\to \infty} a_n b_n ...
3
votes
2answers
56 views

Find the limit $\lim_{n\to \infty} k_n/2^n$ for $k_1=0$ and $k_{n+1}=k_n+\sqrt{1+k_n^2}$

$k_n$ is defined with $k_1=0$ and $k_{n+1}=k_n+\sqrt{1+k_n^2}$. This is homework, please do not provide a complete solution edit : One of the many things I tried is to make it the root of an equation ...
1
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6answers
1k views

Looking for the logic of a sequence from convolution of probability distributions

I am trying to detect a pattern in the followin sequence from convolution of a probability distribution (removing the scaling constant $\frac{6 \sqrt{3}}{\pi }$: ...
0
votes
1answer
29 views

Need help identifiying identity to use

$$\sum_{i=0}^n\frac{{n \choose i}(-1)^n}{i+1}$$ I rearranged terms so that I get $$n!\sum_{i=0}^n\frac{(-1)^n}{(i+1)!(n-i)!}$$ and then looked at the partial sums but I am not able to get anywhere ...
0
votes
2answers
101 views

Sum of sequence of cubes and summation on the upper index

Express the sum of the sequence of cubes as a polynomial in n using the summation on the upper index formula: $$ \sum\limits_{k=0}^n\binom{k}{m} = \binom{n+1}{m+1} $$ It has been proven that the sum ...
2
votes
1answer
49 views

Convergence of $a_n=d(u_n,\Bbb{Z})$ where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$.

I would like to study the sequence defined by $$ a_n=d(u_n,\Bbb{Z}). $$ Where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$ and $d(u_n,\Bbb{Z})=\inf\{d(u_n,x):x\in\Bbb{Z}\}$. I do not really ...
1
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0answers
29 views

Convergence of a series: TV show problem [duplicate]

I came across the following video on youtube where a kid was asked to show that $\sum_{n = 1}^{\infty} \frac{\sin{(2n)}}{1 + \cos^4{n}}$ is convergent. He tried to use the integral test but wasn't ...
35
votes
5answers
780 views

Computing $\lim\limits_{n\to\infty} \Big(\sum\limits_{i = 1}^n \sum\limits_{j = 1}^n \frac1{i^2+j^2}-\frac{\pi}{2} \log(n)\Big)$.

In the chatroom we discussed about the asymptotic of $\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}$, and if we think of the inverse tangent integral, it's easy to see that ...
2
votes
1answer
66 views

Simple proof that this sequence converges [verification]

This is a relatively simple problem. I'm just making sure I have the right idea here. I'd like to prove that the sequence $\displaystyle a_n = 1 + \frac{1}{n^{1/3}}$ converges. My proof is: We ...
0
votes
1answer
37 views

question involving power series for ln(x+a)

please could you help with this question. If a and b are small compared with x, show that $$ln(x+a) - lnx = \frac{a}{b}(1 + \frac{b-a}{2x})(ln(x+b) - lnx)$$ I've tried expanding ln(x+a) as a taylor ...
0
votes
1answer
56 views

Integral Test for convergence of a series

"Consider the series given by $$\sum_{n=2}^{+\infty}\frac{1}{n\ln n(\ln(\ln n))^{\alpha}}$$ for $\alpha>1$. Use the Integral Testo to conclude if the series is convergent or not." I tried to make ...
2
votes
2answers
77 views

How can I prove that $\sum_{k=1}^{\infty}\frac{(2\sqrt[k]k-1)^k}{k^4}$ is convergent?

I'm trying to prove that the sum $\sum_{k=1}^{\infty}\frac{(2\sqrt[k]k-1)^k}{k^4}$ is convergent. I've tried Cauchy's root test - but I get the limit to be 1, so the test is inconclusive. I also ...
2
votes
2answers
58 views

Select a subsequence to obtain a convergent series.

Does there exists strictly increasing sequence $\{a_k\}_{k\in\mathbb N}\subset\mathbb N$, such that $$ \sum_{k=1}^{\infty}\frac{1}{(\log a_k)^{1+\delta}}\lt \infty, $$ where $\delta>0$ given and ...
2
votes
1answer
177 views

Simplifying big expression

What to do with this? $$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$ Can it be simplified?
6
votes
2answers
154 views

Convergent or divergent $\sum\limits_{n=0}^{\infty }{\frac{1\cdot 3\cdot 5…(2n-1)}{2\cdot 4\cdot 6…(2n+2)}}$

\begin{align} & \sum\limits_{n=0}^{\infty }{\frac{1\cdot 3\cdot 5...(2n-1)}{2\cdot 4\cdot 6...(2n+2)}} \\ & \text{ordering} \\ & a_{n}=\frac{1\cdot 3\cdot 5...(2n-1)}{2\cdot 4\cdot ...
1
vote
1answer
36 views

How to find the solution of integer equation group

I have the following problem: to find the general item of the following equation: let $a_1=b_1=1$, $$a_{n+1}=6a_n+2b_n, b_{n+1}=3a_n+2b_n$$ for any $n\geq 1$, find $a_n=?, b_n=?$
-1
votes
3answers
40 views

Determine whether this series converges using ratio test $\sum_{n=1}^{\infty}\prod_{j=1}^{n}\frac{3j-1}{4j-3}$ [closed]

Umgh... I just have no idea, srsly, I've never done anything with sum of products
1
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4answers
53 views

how do I solve this arithmetic series

I have this arithmetic series $3+7+11+...+35+39$ to solve. So I see that there is a difference by 4 between the numbers and that there is a total of 9 terms. I plug these values in the following ...
5
votes
4answers
359 views

Does There Exist an Explicit Formula Describing Every Possible Sequence of Numbers?

This thread was previously titled "Does a Set Require an Explicit Formula to Exist?". I'm reading H. Enderton's Elements of Set Theory and working through understanding the Zermelo-Fraenkel axioms. ...
2
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1answer
452 views

Simplify a summation to reduce computation time

I am working on an optimization problem in which the following summation should be calculated in a computer program over a billion times. Therefore, I am looking for the possibility of somewhat ...
1
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1answer
50 views

test the convergence of an infinite series

How to prove that the $\displaystyle \sum_{n=1}^{+\infty} (1-e^{(-1/n^2)})$ series is convergent? I can not find a number to use the comparison test!
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votes
1answer
63 views

Proving that a sequence is Cauchy on the basis of squeeze theorem

Let $\{x_n\}$ be a sequence of real numbers such that $$|x_n| \leq \frac{2n^2 + 3}{n^3 + 5n^2 +3n + 1}$$ Prove $\{x_n\}$ is a Cauchy sequence Proof: Suppose that ${x_n}$ be a sequence of real ...
3
votes
2answers
91 views

Convergence of a crazy power series

"Let $\alpha$ be a given real number, $\alpha>0$ and $\alpha \notin \Bbb{N}$. Proof that the series $$\sum_{n=1}^{+\infty}\frac{\alpha(\alpha-1)(\alpha-2)\dots(\alpha-(n-1))}{n!}x^n$$ converges for ...
1
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2answers
61 views

Conditions for convergence of $\sum\limits_{n=1}^\infty{a^nf(n)}$

assume $a>0$, and for all $n$ we have $0 \leq f(n) \leq 1$. Is there a necessary and sufficient condition on the series $f(n)$ for which $\sum\limits_{n=1}^\infty{a^nf(n)}<\infty$ ? Thanks!
1
vote
0answers
100 views

Prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent.

I need to prove that $\{\sqrt[n]{e^{n+1}}\}$ is convergent, and find its limit using this theorem: Let $f:E \to R$ with $x_{0} \in E$ an accumulation point of $E$. Then these are equivalent: 1)$f$ ...
1
vote
2answers
712 views

Summation of series involving logarithm: $\sum (n+2)\ln 2^n$

The following question is: Show that $\sum\limits_{r = 1}^n {r(r + 2)} ={n \over 6}(n+1)(2n+7).$ Using this results, or otherwise, find, in terms of $n$, the sum of the series ...
1
vote
4answers
114 views

$\text{Prove that}$ $\frac{\sin(\frac{n+1}2)*\cos(\frac n2)}{\sin\frac 12} \ge\frac n2$

Prove that$$\frac{\sin\left(\frac{n+1}2\right)\times\cos\left(\frac n2\right)}{\sin\left(\frac 12\right)} \ge\frac n2$$ So far I've switched up the problem and gotten it down to all sin functions. I ...
0
votes
1answer
52 views

Is $n^3$ Bounded for $n\in\mathbb{N}$ as ${n\to\infty}$

Is $n^3$ Bounded for $n\in\mathbb{N}$ as ${n\to\infty}$ ? I am wondering if this is the case because I am thinking it bounded below by 1 but my textbook says its unbounded. Can someone explain why?
-1
votes
2answers
50 views

If $x_n \to +\infty$ then $(1+1/x_n)^{x_n} \to e$

I want to prove that if $x_n$ is a numerical sequence such that $\lim_n x_n=+\infty$ then $\lim_n (1+1/x_n)^{x_n}=e$. Should I pass by the continuous case (studying $f(x)=(1+1/x)^x$ for $x \to ...
6
votes
2answers
161 views

Does the series converge or diverge?

I want to check, whether $$\sum\limits_{n=0}^{\infty }{\frac{n!}{(a+1)(a+2)...(a+n)}}$$ converges or diverges. $a$ is a constant number Ratio test $$\begin{align} & ...
0
votes
1answer
41 views

If $\sum _{n=0 } ^{\infty } a _n < \infty$ $\sum_{n =0 } ^{\infty } b _n < \infty$ then $\sum_{n=0 } ^{\infty } a _nb _n < \infty$,

This seem obvious to be true but I'm unsure how to prove it or if there ara basic results about inifnite sums that apply. If $\sum _{n=0 } ^{\infty } a _n < \infty$ $\sum_{n =0 } ^{\infty } b _n ...