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

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0
votes
1answer
17 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
votes
2answers
88 views

How do I show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m))) + \cdots$ converges for all real numbers $m$?

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
24 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
vote
1answer
16 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
0answers
9 views

Calculate the order of error for the (summed) Midpoint rule?

I'm reading a comparison of the summed rectangle and and midpoint rules for estimating the value of an integral. The midpoint rule: $\displaystyle\int_a^b \! f(t) \, \mathrm{d}t \simeq f(a + h/2)h$ ...
0
votes
1answer
17 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
vote
1answer
16 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
vote
3answers
66 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 $n!^{1/n}>\ln(n)$, ...
5
votes
1answer
68 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 $\frac{1}{2}$, but how to prove it using integral or Euler sum only (without ...
1
vote
1answer
50 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
31 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\}\\ ...
1
vote
2answers
54 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
40 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, ...
0
votes
1answer
22 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
45 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 ...
0
votes
5answers
776 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
25 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
47 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
34 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
vote
0answers
27 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 ...
15
votes
1answer
118 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
46 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
23 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
35 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 ...
1
vote
2answers
54 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 ...
-1
votes
0answers
33 views

Power series for $\ln(1+x)$ and an estimate for $\ln(b/a)$ when $b\approx a$ [on hold]

I'm stuck on this question involving the power series for ln(1+x): $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ...$$ "Apply Maclaurin's series to establish a series for ln(1+x). ...
2
votes
2answers
43 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
159 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?
4
votes
2answers
109 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
25 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
30 views

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

Umgh... I just have no idea, srsly, I've never done anything with sum of products
0
votes
4answers
31 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 ...
4
votes
4answers
198 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
votes
1answer
38 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
vote
1answer
40 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!
-1
votes
1answer
27 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 ...
2
votes
1answer
44 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
vote
2answers
43 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
80 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)$f$ ...
0
votes
1answer
30 views

summation of series with weird general term [on hold]

I'm having problem with this question: Find the sum of (n+1)2n+(n+2)(2n-1)+(n+3)(2n-2)+...+(2n-1)(n+2)+2n(n+1) ps: if possible, leave some advice for finding general term when dealing with summation ...
0
votes
0answers
30 views

What sequence should I use to visit color wheel? [on hold]

I am writing a program where I need to color N objects with unique colors. I have set up a function that maps the floating point value k [0,1] to a smoothly varying rainbow (say rgb=wheel(k)). If N ...
1
vote
2answers
55 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
98 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
50 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
40 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 ...
-1
votes
2answers
42 views

What's the best way to find the sum of this sequence? [on hold]

I've got the following sequence: $$ 160-157+154-151+148-145+...+4-1 $$ How to find a sum of it?
6
votes
2answers
133 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
40 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 ...
2
votes
2answers
36 views

summation of series (odd and even case)

Can anyone please answer this for me,it involves alternate signs which is different from normal summation formula. Q:Find the sum to n terms of the series $ 1^3 - 2^3 + 3^3 - 4^3 + \ldots -(n-2)^3 + ...
3
votes
2answers
123 views

Sums $\sum_{n=1}^{N}\sqrt{4n+1}$

I need to find sum of the first N terms of the sequence whose nth term is as follow : T(n)= $\sqrt{4*n+1}$ So the sequence is : $\sqrt{5}$,$\sqrt{9}$,$\sqrt{13}$,$\sqrt{17}$,$\sqrt{21}$...... ...