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

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1
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2answers
26 views

$\lim_{n \to\infty} p_n = p$ implies $\lim_{n \to \infty}p_n^3 = p^3$

In an example from my lecture notes, I have that if $( p_n )_n$ is a sequence and $\displaystyle\lim_{n \to \infty} p_n = p$, then $\displaystyle\lim_{n \to \infty} p_n^3 = p^3$. I don't understand ...
1
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2answers
42 views

Series question,related to telescopic series, 1/2*4+ 1*3/2*4*6+ 1*3*5/2*4*6*8 …infinity

The series is $$\frac{1}{2*4}+ \frac{1*3}{2*4*6}+ \frac{1*3*5}{2*4*6*8}+....$$ It continues to infinity.I tried multiplying with $2$ and dividing each term by$(3-1)$,$(5-3)$ etc,starting from the ...
2
votes
1answer
65 views

Recursive sequence $a_{n+1} = \sqrt{a_1 + … + a_n}$

I need a hint to solve this: Let $a_1 = 1$ and define a sequence recursively by $$a_{n+1} = \sqrt{a_1 + a_2 + ... a_n}$$ Show that $$\lim_{n \to \infty} \dfrac{a_n}{n} = \dfrac{1}{2}$$ Any help? ...
1
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1answer
18 views

Train distances leaving at certain times

A train leaves Boston to Fort Lauderdale traveling at $125$ mph. An hour later, another train leaves Fort Lauderdale traveling to Boston at a rate of $140$ mph. When the two trains meet each other, ...
0
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1answer
40 views

$a_{2014}$ sought in a sequence

$a_n$ sequence is defined as follows: $a_1=0$ and $a_n$ is the smallest positive whole number, such that within $a_1,a_2,...,a_{n-1},a_n$ there is no arithmetic progression of 3 terms. How do I ...
0
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1answer
59 views

What comes next? For 8 year olds

This question is from the homework of my niece. She is 8 years old. And I could not help her with this question. There are 5 x 3 cells. And there is a number in each cell. Problem asks what should be ...
0
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2answers
29 views

Does a Sequence Converge to 0 if and only its Reciprocal Sequence Diverges to Infinity?

I was considering yesterday whether or not the question in the title is, in fact, true. I believe that the definition of a convergent sequence is $(\forall \epsilon>0)(\exists ...
-2
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2answers
37 views

Limit of $n \sum_{k=1}^{2n+1} \frac{1}{n^2+k}$ [on hold]

How can I find $$\lim_{n\to +\infty} n \sum_{k=1}^{2n+1} \frac{1}{n^2+k} ?$$
1
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1answer
36 views

Can one express in a compact form a double, triple, etc., alternating series?

Trivially, a series like$\ a_0-a_1+a_2-a_3+...$ can be written as$$\ \sum_{i=0}^\infty (-1)^ia_i.$$ But what if I want to rewrite$\ a_0+a_1-a_2-a_3+a_4+a_5-...$ ,$\ a_0+a_1+a_2-a_3-a_4-a_5+...$ and so ...
0
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2answers
27 views

Rewrite $\sum_{n\ge0}{z^{kn}}$ to $\sum_{n\ge0}{f(n,k)z^n}$

In the simplest case: $\sum_{n\ge0}{z^{2n}}=\sum_{n\ge0}{\frac12((-1)^n+1)z^n}$. How to express $f(n,k)$ in closed form? If it's intractable, how to avoid piecewise expression?
1
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1answer
41 views

Closed form for $\sum _{k=4}^{\infty }{\frac { \left( -\ln \left( 2 \right) \right) ^ {k}\zeta \left( 4-k \right) }{k!}}$

Can anybody find a closed form for this infinite sum? $$S = \sum _{k=4}^{\infty }{\frac { \left( -\ln \left( 2 \right) \right) ^ {k}\zeta \left( 4-k \right) }{k!}},$$ where $\zeta$ is the Riemann ...
1
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2answers
30 views

Partial sum formula of a polynomial series?

I am trying to find the partial sum formula of the following series: $$ \sum_{y=1}^{\infty} \frac{4y^2-12y+9}{(y+3)(y+2)(y+1)y} $$ I have tried using Faulhaber's formula without success. I have also ...
2
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0answers
37 views

Closed form of arctanlog series

What tools would you recommend me for $$\operatorname{arctan}\left( \frac{1}{1}\right)\log\left(1+\frac{1}{1}\right)+\operatorname{arctan}\left( ...
1
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2answers
50 views

Sequence of alternating $0$'s and $1$'s in terms of $i$?

How to redefine the function $f(n) = \begin{cases} 1, & \text{if $n$ is even} \\ 0, & \text{if $n$ is odd} \end{cases}$ in terms of arithmetic operations using ⅈ?
0
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0answers
34 views

Resources to investigate rational numbers

I have been told that resources like Mathematica's Number Recognition (which I've never tried myself) and the Inverse Symbolic Calculator (ISC) can be used to find possible closed forms for real ...
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0answers
32 views

prove that a set is non empty

Given that $D(k)$ is an increasing sequence and $D\overset{k}{\rightarrow} A$, where $A$ is a constant real. We have to show that the set $$\{k \mid D \leq Q < A\}$$ is non empty? where $Q$ is a ...
0
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1answer
25 views

Proving uniformly convergence on a Banach Space

Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$ and $$ {\cal L}_0^2(\mathbb R)=\left\{f:\mathbb R\mapsto\mathbb R\ |\ ...
2
votes
1answer
54 views

Find the closed form of an n-sum

I'd like to find the closed form or a quickly converging rewriting of the following n-sum: ...
1
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3answers
36 views

Proving that the sequence $\{x^2 (\cos (1/x) - 1)\}$ does not converge to $0.$

I'm trying to prove whether or not $$ \sum_{x = 1}^\infty x^2 \left (\cos\left (\frac{1}{x}\right ) -1 \right ) $$ converges. Based on graphs, I think that the sequence $\{x^2 (\cos (1/x) - 1)\}$ ...
1
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1answer
36 views

Any necessary and sufficient condition for $\displaystyle\lim_{n \to \infty}\frac{a_n}{b_n}=0$?

I am looking for any established result that is necessary and sufficient for $\displaystyle\lim_{n \to \infty}\frac{a_n}{b_n}=0$ for any real sequences $\{a_n\} \text{and} \{b_n\}\ne 0$, where both ...
0
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0answers
10 views

Show Cantor function is continuous using sequences

To construct the Cantor set, $C$: Define $J_{1,1}:= (1/3,2/3)$. And define $J_{2,1}:= (1/9, 2/9)$ and $J_{2,2}:= (7/9,8/9)$. Continue to get sets $J_{n,k}$ and define: $K_1 = [0,1] \setminus ...
2
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2answers
13 views

Numbering fields in a growing square for easy calculation based on coordinates

I have a square that slowly grows into a bigger square by adding more and more copies of the original square as shown in this picture below: It starts with one square $(0)$, and then grows by 3 ...
1
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0answers
16 views

Which $L \subset [0,1]$ equal the set of limits of a sequence of a sequence in $[0,1] \setminus L$?

I was glanced at this question here and it cause me to wonder the following: Question: Is there a simple description of the subsets $L \subset [0,1]$ with the property that there exists a sequence ...
0
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4answers
21 views

Why a sequence $\{x_k\}$ in $S$ such that $||x||>k$ cannot be a convergent subsequence?

This is an excerpt from my text: A set $S \subset \mathbb{R}^n$ is said to be compact if for all sequences of points $\{x_k\}$ such that $x_k\in S$ for each $k$, there exists a subsequence ...
0
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1answer
36 views

Solving recurrence relation in form of $f(n)=f(n-1)+K-n$

I asked a question (now deleted it) on finding a relation between $$4,10,15,\cdots$$ I studied a a little about recurrence relation and solving them. for the above sequence I observed that $$f(n) = ...
2
votes
2answers
56 views

Does $\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k \ln(k)^2}$ converge?

Does $$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k \ln(k)^2}$$ converge over $\Bbb R$, and if so, towards which limit?
0
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3answers
72 views

Can I know the value of an infinite serie?

$$\sum\limits_{n=0}^{\infty}\frac{n}{e^n}$$ I have found through a software that the value is $\dfrac{e}{(e-1)^2}$. I've been trying to do it manually but I am getting $\dfrac{\infty}{\infty}$, ...
0
votes
1answer
31 views

If $f_{n-1}(x)=nf_{n}'(x)$ , write $ f_n(x)$ as a function of $f_1(x)$

Let $f_{n-1}(x)=nf_{n}'(x)$ for all $n>1$, and $f_1(x)=a$, what is the expression depicting the relationship between $ f_n(x)$ and $f_1(x)$? I need help with this series.
0
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0answers
23 views

If $y_m \to y$ in $H$ then $|y_m|_H \leq C|y|$ for this sequence?

Let $w_i$ be a basis for a Banach space $V$. We have $V \subset H$ a continuous and dense embedding into a Hilbert space $H$. Define $y_m = \sum_{i=1}^m a_{im}w_i$. We have that $y_m \to y$ in a $H$ ...
6
votes
3answers
83 views

Example of an $(a_n)$ sequence with exactly $k$ limit points

It is a well-known result that the sequence $$ a_n= \frac{(-1)^nn}{n+1}, $$ has two limit points, and these are $1$ and $-1$. I'm just looking for some examples of sequences that have exactly $k$ ...
0
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1answer
50 views

Does the series $\sum_{n=1}^{\infty} \sin\left((-1)^{n}\over n\right) $ converges conditionally? [on hold]

Which convergence tests can I use in order to show that the series $$\sum_{n=1}^{\infty} \sin\left((-1)^{n}\over n\right) $$ converges conditionally.
7
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1answer
86 views

A cute limit $\lim_{m\to\infty}\left(\left(\sum_{n=1}^{m}\frac{1}{n}\sum_{k=1}^{n-1}\frac{(-1)^k}{k}\right)+\log(2)H_m\right)$

I'm sure that for many of you this is a limit pretty easy to compute, but my concern here is a bit different, and I'd like to know if I can nicely compute it without using special functions. Do you ...
3
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0answers
30 views

Explicit definition from recursive definition

I have the recursive definition: $$a_0=0,~a_{n+1} = 28 + a_n + \left\lfloor\frac{a_n}{16}\right\rfloor$$ I want to create an explicit form for that. I was able to transform the problem into finding an ...
0
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2answers
28 views

Number Sequences

These are the terms the question gave me. Term 1 = 1 Term 2 = 1 Term 3 = 2 Term 4 = 3 Term 5 = 5 Term 6 = 8 Term 7 = 13 Term n = ? I found out the pattern which is Term 2 is Term 1 + 0. Term 3 is ...
2
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2answers
31 views

Another Summation

After looking at the question here Computing summation I wondered if it might be possible to evaluate the following summation with a similar-looking summand term but with $2n$ instead of $2^n$: ...
0
votes
1answer
61 views

Prove (or disprove) this $\sum_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergence? [on hold]

Let $a_{n}>0$ be a sequence, and $0<a\le 1$, such that $\sum\limits_{n=1}^{\infty}(a_{n})^a$ converges. Prove or disprove $\sum\limits_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergent. I ...
0
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1answer
20 views

Convergence of sequence of functions on Banach space

Let $\{f_{\alpha_n}\}\subset{\cal L}_2^0(\mathbb R)$ be a sequence function converging to $g$ where ${\cal L}_2^0(\mathbb R)$ is a Banach space defined by $$ {\cal L}_2^0(\mathbb ...
1
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2answers
40 views

Convergence of $\sum_{n} \frac{a_n}{2^n}$ where $a_n\rightarrow +\infty$

Does there exists a sequence $a_n$ of positive reals such that $a_n\rightarrow \infty$ but $\sum_{n}\frac{a_n}{2^n}$ converges? (I considered $a_n=n$, but couldn't prove convergence or divergence of ...
2
votes
3answers
68 views

$\sum (\frac{1-2n}{6+2n})^n $ converges?

Verify if $$\sum_{n=0}^{\infty} \left(\frac{1-2n}{6+2n}\right)^n $$ converges The root test is inconclusive and the limit of the general term is 0. I think I should use the comparison test, in this ...
0
votes
0answers
42 views

Summation of infinite series split into two sums [on hold]

Does $$\sum_{r=1}^\infty (r^2+r) = (\sum_{r=1}^\infty r^2) + (\sum_{r=1}^\infty r)$$ and if so, what is the proof of this? Does the sum of (1/(r^2) + 1/r^3) from 1 to infinity = sum of 1/r^2 from 1 ...
0
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0answers
17 views

Complex form of Fourier series - help

Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ $$a_0=\frac{2}{b-a}\int_{a}^{b}f(t)dt,$$ ...
2
votes
3answers
71 views

If $a_{n+1}=a_n+\frac1{a_n}$, then $a_n/n$ converges to $0$

Let $a_{n+1}=a_n+\dfrac1{a_n}$, with $a_n=1$. Prove $\lim \limits_{n\to \infty }\left(\dfrac{a_n}{n}\right)=0$. Now I already know that it is monotonically increasing and that $a_n\to \infty$ as ...
3
votes
2answers
42 views

Fractional Sobolev space $H^{1/2}(-\pi,\pi)$

Let $H^{1/2}(-\pi,\pi)$ be the space of $L^2$ functions whose Fourier series coefficients $\{c_n\}_n$ satisfy the summability constraint $\sum_n |n| |c_n|^2 < \infty$. Are functions in ...
0
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0answers
38 views

Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$ Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that $$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le ...
3
votes
2answers
74 views

Convergence of $\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$

Does the series $$\sum \frac{(2n)!}{n!n!}\frac{1}{4^n}$$ converges? My attempt: Since the ratio test is inconclusive, my idea is to use the Stirling Approximation for n! $$\frac{(2n)!}{n!n!4^n} ...
0
votes
0answers
28 views

The value of $\sum^{100}_{r=2} \frac{3^r (2-2r)}{(r+1)(r+2)} =$

Problem : The value of $$\sum^{100}_{r=2} \frac{3^r (2-2r)}{(r+1)(r+2)} =$$ Solution : $$\sum^{100}_{r=2} \frac{3^r (2-2r)}{(r+1)(r+2)} $$ Putting r =2,3,4 $\cdots$ $$T_1 = \frac{3}{2}-3$$ ...
2
votes
4answers
55 views

If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots $ n terms… [on hold]

Question( from sequences) : If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots $ n terms = $(4000)(4000!)$ Then what is the value of n. How to proceed in this please suggest , will be of great help to ...
4
votes
1answer
60 views

Find the sum to n terms of the series

Find the sum to n terms of the series $$\frac {\sin x}{\cos x+\cos2x} + \frac {\sin2x}{\cos x+\cos4x} + \frac {\sin3x}{\cos x + \cos6x} +\dotsb $$ How can I solve this? Here is what I did for the ...
0
votes
1answer
15 views

Function of a sequence, how to answer these types of questions.

I don't quite get this idea of taking the function of a sequence, and what it implies. The questions I am getting go a bit like this; Given a sequence $s_n$ that converges to some $a$ as $n$ gets ...
1
vote
2answers
37 views

What is the sum $\sum_{r=1}^\infty \frac{r}{4r^4+1}$ equal to?

Problem : If $$T_r =\frac{r}{4r^4+1}$$ then the value of $$\sum^{\infty}_{r=1} T_r$$ is ? How to start such problem I am not getting any clue on this please suggest thanks .