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

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0
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0answers
4 views

Uniform Integrability - different characterisation - prove (ii)

Probability with Martingales: For the 'only if' part assuming the hint is true, then I guess we have $\forall \varepsilon_1 > 0, \exists K \ge 0$ s.t. $$E[|X|1_{|X| > K}] < \...
0
votes
1answer
7 views

Uniform Integrability - sufficient condition and bounded convergence theorem with weaker hypothesis

Probability with Martingales: How does the result follow? Do we choose $K = (\frac{\varepsilon}{A})^{\frac{1}{1-p}}$ Why do we have that inequality?
0
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0answers
4 views

Uniform Integrability - different characterisation - prove hint

Probability with Martingales: For the 'only if' part how to prove the hint? i'm guessing it's something to do with $$E[X 1_F] \le E[X1_{\Omega}]$$ $$= E[X 1_{|X| > K}] + E[X 1_{|X| \le K}]...
1
vote
1answer
31 views

How to evaluate $\sum_{n=1}^{\infty}a_n$?

If $$a_{n}=1-\frac{1}{2}+\frac{1}{3}-\cdots +\frac{\left ( -1 \right )^{n-1}}{n}-\ln 2$$ then how to eveluate $$\sum_{n=1}^{\infty}a_n$$ does it converge?
1
vote
4answers
20 views

Convergence and sum of series with exponents

So the question is how can I see if this series : $$\displaystyle\sum_{n=1}^{\infty} \frac{1}{(4-(-1)^n)^n}$$ converges and find its sum. So I would probably need to use the Leibnitz criterion for ...
2
votes
2answers
72 views

Does this alternating sum of roots converge to $\sqrt2$?

This problem arose from what I'm hesitant to call an investigation into a certain type of "quadrature". Starting with the unit disk, I partition it into $p$ pieces by cutting the disk with vertical ...
1
vote
5answers
60 views

how to determine if this series converges?

I was trying to find out if the series: $$\sum^{\infty}_{n=1}n^3e^{-n} $$ converges. I tried applying the Cauchy test, $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{n^3e^{-n}}=\...
5
votes
1answer
43 views

Prove the uniform convergence of $f_{1}(x)= \sqrt x , f_{n+1}(x)=\sqrt{x+f_n(x)}$ in $[0,\infty]$

As far as I understand most of these questions use the M-test, but I can't find a series that suffices.
2
votes
2answers
44 views

How to find the N-th 3 word sequence within the following constraints

I have a list of words. Let's say that I have an algorithm(explained below) to generate the permutations in a specific order. I want to be able to find the N-th permutation easily. I want to make ...
1
vote
1answer
53 views
+150

Is it possible to find the partial sum?

Let, $a_n=\frac {3^{n+1}}{1+2^{n+1}},n\geq0.$ Let $S_n$ be the partial sum defined by $$S_n=\sum_{i=0}^{n}a_i.$$ Is it possible to write a closed formula for $S_n.$ I have no idea how to do this. Any ...
0
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0answers
20 views

Difficulties understand the series solution of $(1-x^4)y''-8x^3-12x^2y=0$

Solve: $(1-x^4)y''-8x^3-12x^2y=0$ using the solution: $$y=\sum_{n=0}^{\infty}a_nx^n$$ Let's differentiate y: $$y'=\sum_{n=1}^{\infty}(n)a_nx^{n-1}$$ $$y''=\sum_{n=2}^{\infty}(n)(n-1)a_nx^{n-2}$$ ...
2
votes
0answers
28 views

Existence of sequences

Given real numbers $a, b, c$ such that $a^2= b^2+c^2$, there exists three sequences of natural numbers $a_n, b_n, c_n$ such that $a_n(a_n+1)= b_n(b_n+1)+c_n(c_n+1)$. The ratios $b_n/a_n$ and $ c_n/a_n$...
3
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5answers
232 views

Sum of real powers: $\sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$

Let $\{x_i\}_{i=1}^{N}$ be positive real numbers and $\beta \in \mathbb{R}$. Can we say that: $$ \sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$$ I know that this holds if $...
0
votes
1answer
15 views

Three negative and three positive $1$ s in a serie(updated)

I asked the same question here:Three negative and three positive $1$s in a serie But when I see that it was closed because of being unclear I decided to make It better and ask again. First I want to ...
0
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1answer
31 views

If $a_n$ converges to $A$ and a is a non-empty set, show that $4A^2=A^2+4+4/(A^2)$ [on hold]

Consider the example of the sequence defined recursively by $$a_{n+1} = \frac12\left(a_n+\frac{2}{a_n}\right)$$ for $n\geq1$. By easy algebra,$$4a^2_{n+1} = a^2_n+4+\frac{4}{a^2_n}$$ Suppose ...
0
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1answer
45 views

Prove that $\sum_{n=1}^{\infty }\frac{B_{2n}}{(2n-1)!}=\frac{1}{2}-\frac{1}{(e-1)^2}$

Prove that $$\sum_{n=1}^{\infty }\frac{B_{2n}}{(2n-1)!}=\frac{1}{2}-\frac{1}{(e-1)^2}$$ My idea is to find the Taylor series of $\frac{1}{(e^x-1)^2}$, but it seems not useful. Any helps, thanks
0
votes
3answers
89 views

Which function of $x$, other than $x +c$, and Integral of ($\cos x)^2+(\sin x)^2$ have derivative =$1$

It is a simple question:Which function of x, other than x +c, and Integral of (cosx)^2+/(sinx)^2 have derivative =1. Alternative question, Which other equation gives its derivative as a real number. ...
35
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2answers
945 views
+50

Prove that $\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

Question. Let $$ f(x)=\!\left\{\,\,\, \begin{array}{ccc} \displaystyle{\left\lfloor{1\over x}\right\rfloor}^{-1}_{\hphantom{|_|}}&\text{if} & 0\lt x\le 1, \\ & \\ 0^{\hphantom{|^|}} &\...
1
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2answers
36 views

Complex series should sum to zero but it's a puzzle

If we have a finite sum defined as $$\frac{1}{N}\sum\limits_{n=N/4}^{3N/4-1} e^{-4\pi ink/N}$$ (where $k$ is an integer and $N$ is divisible by $4$), then how can we show that this sum is equal to $...
1
vote
1answer
31 views

Sum of a series with exponential and polynomial terms

I have reduced the expression that I am working on to the following sum of series, which is definitely converging. It would be great if someone can help me out with this or suggest ways this can be ...
-1
votes
2answers
28 views

Compute the limit of the sequence given by bn =(1+(3.4/n))^n [on hold]

If a sequence $c_1$ , $c_2$ , $c_3$ ,... $c_{n-1}$, $c_n$, $c_{n+1}$,... has limit $K$ then the sequence $e^{c_1}$, $e^{c_2}$, $e^{c_3}$ , ... $e^{c_{n-1}}$, $e^{c_n}$, $e^{c_{n+1}}$,... has limit $...
2
votes
2answers
60 views

Recurrence relation solution $a_{n}=\frac{a_{n-1}}{a_{n-1}-a_{n-2}}$

I want to find the analytic form of the recurrence relation $$a_{n}=\frac{a_{n-1}}{a_{n-1}-a_{n-2}},\,a_{1}=2,\,a_{2}=1 $$ but when looking at the results they seem chaotic. Is it possible that it ...
6
votes
3answers
117 views
+50

A System of Infinite Linear Equations

Suppose that $\{a_{i}\}_{i=-\infty}^{\infty}$ with $\sum_{i=-\infty}^\infty a_{i} \lt \infty$ is known and that $\{b_i\}_{i=-\infty}^{\infty}$ is such that $$\sum_{i=-\infty}^\infty a_{i}b_{-i} =1,$$ ...
13
votes
2answers
237 views

Daunting series of integrals: $\sum_{n=2}^\infty\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log(\frac{1-\sin x}{1+\sin x})dx$

My coleague showed me the following integral yesterday \begin{equation} I=\sum_{n=2}^{\infty}\int_0^{\pi/2}\sqrt{\frac{(1-\sin x)^{n-2}}{(1+\sin x)^{n+2}}}\log\left(\!\frac{1-\sin x}{1+\sin x}\!\...
7
votes
2answers
282 views

Limit of $\int_0^1\frac1x B_{2n+1}\left(\left\{\frac1x\right\}\right)dx$

Set $$u_n= \int_0^1 \frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx\tag1$$ where $\left\{t\right\}=t-\lfloor t \rfloor$ denotes the fractional part of $t$ and where $B_n(\cdot)$ are the ...
3
votes
1answer
52 views

Elementary proof for non-existence of a pointwise convergent subsequence of $\{\sin (nx)\}$

My teacher showed this proof using the dominated convergence theorem or Fourier analysis, but I wonder if there is an elementary proof of this problem. My teacher said it is difficult to solve this in ...
0
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1answer
28 views

Series - calculating the sum

I need to analyze the series $\sum_{n=0}^\infty{x^n\cos(\frac{n\pi}{2})}$, and I have already shown that it converges. Now I am trying to find the sum, and I noticed that $\cos(\frac{n\pi}{2})$ gives ...
1
vote
1answer
49 views

Doubt on the comparison test: can I still evaluate $\lim \limits_{n \to \infty} \frac{a_{n}}{b_{n}}$ if $b_{n}$ might be $0$ for some $n$?

Suppose that $(b_{n})_{n \in \mathbb{N}}$ is a sequence which is not identically equal to $0$ (but which may have elements equal to $0$). Suppose also that I know that $\sum b_{n}$ converges ...
0
votes
0answers
17 views

Closed-form expression for recursive function $F(n,a)=F(n-1,a)+F(n-(1+a),a), F(b,a)=1, b={1,2,3…,a+1}$

For the most common cases, $a=0$ and $a=1$, the explicit solutions are generally known as: $$F(n,0)=2^{n-1}$$ $$F(n,1)=\dfrac{\phi^n-(-\phi)^{-n}}{\sqrt{5}}$$ Is it possible to derive general closed-...
0
votes
4answers
83 views

How to show convergence and evaluate $\sum\limits_{n=0}^{\infty} \frac{n}{(n+1)!}$ [duplicate]

I try to evaluate series ${n\over(n+1)!}$ obviously the term $s_n = {n\over (n+1)!} = {1\over (n-1)! (n+1)}$ converges, which has a limit = $0$ when $n\to \infty$ .Then I was stuck and I don't know ...
1
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2answers
60 views

Sides of the triangles are in G.P.

Question:- The sides of a triangle are in G.P. and it's largest angle is twice the smallest one. Prove that the common ratio of the G.P. lies in the interval $(1,\sqrt{2})$ Attempt at a solution:- ...
1
vote
1answer
18 views

Proof radius of convergence for zero and infinite (power series)

I deleted my last question because there was a huge mistake inside. Given: $R$ is the radius of convergence of $\sum_{n=0}^{\infty} a_{n}x^{n}$, also suppose that $\lim_{n\rightarrow \infty} \left | \...
2
votes
1answer
105 views

Converting from Closed Form

Let $A(n) = \lfloor n/2+\log_2(n)-\log_2(2) \rfloor$. Is there an easy way to convert this closed form into a recursive form? If so, what is the general method, and how might it be applied here. If ...
33
votes
1answer
671 views

Closed form for $\left(1+\left(\frac{1}{2}+\left(\frac{1}{3}+\left(\frac{1}{4}+\cdots\right)^2\right)^2\right)^2\right)^2$?

Nested squares seem to be more promising than nested radicals, since they give rational approximations and in principle can be expanded into a series. These two expressions converge numerically: $$\...
3
votes
2answers
318 views

Finding the general formula of a sequence: $3,8,23,68,203,608,\cdots$

I have the following sequence : $$3,8,23,68,203,608,\cdots$$ I have found that definition by recurrence of this is $$a(n)=3a(n-1)-1$$ where $a_0=3$ as the first term. I want to find the explicit ...
2
votes
0answers
36 views

Convergence of a series of complex numbers.

Let $f : \mathbb C \to \mathbb C$ be a non constant entire function. Does the series $\sum_{n=1}^ \infty \frac{1}{n} f(\frac {z}{n})$ converges at any point $z \in \mathbb C$ ? I think this will not ...
26
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2answers
696 views

Show that $\sum_{n=0}^{\infty}\frac{2^n(5n^5+5n^4+5n^3+5n^2-9n+9)}{(2n+1)(2n+2)(2n+3){2n\choose n}}=\frac{9\pi^2}{8}$

I don't how prove this series and I have try look through maths world and Wikipedia on sum for help but no use at all, so please help me to prove this series. How to show that $$\sum_{n=0}^{\...
-1
votes
1answer
26 views

Limits superior: $\limsup_{n\rightarrow\infty}x_k=\sup\{\lim_{k\rightarrow\infty}x_{n_k}\}$?

Let $x\in \ell^\infty$. By definition we have $\limsup_{n\rightarrow\infty}x_n=\lim_{n\to\infty}\sup_{m\geq n} x_m$. Can you also write $\limsup_{n\rightarrow\infty}x_k=\sup\{\lim_{k\rightarrow\infty}...
3
votes
3answers
90 views

Suppose $\sum a_n $ converges. Does $\sum 2^{-n} a_n $ always converge?

Suppose $\sum a_n $ converges. Does $\sum 2^{-n} a_n $ always converge ? If $a_n$ is non-negative series then the answer is yes. What about the general series?
0
votes
1answer
33 views

What vectors can be generated by permuting and halving?

$x$ is a vector in the unit simplex in $\mathbb{R}^n$, i.e: $$x = (x_1,\dots,x_n)\,\,\,\,\,\,\,\,;\,\,\,\,\forall i: x_i\geq 0\,\,\,\,;\,\,\,\,\,\,\,\,\sum_{i=1}^n x_i = 1$$ Initially, $x=(0,0,\dots,0,...
2
votes
1answer
31 views

Upper bound for $\sum_{i=1}^{N}{x_i^{\beta_i}}$

$\{x_i\}_{i=1}^{N}$ a sequence of positive real numbers and $\{\beta_i\}_{i=1}^{N} $ are real numbers such that $\underset{1 \leq i \leq N}\min{\beta_i}$ > 1. Is it possible to find an upper bound of ...
0
votes
1answer
38 views

Hofstadter's Male-Female Sequences: A Curiosity

I have a curiosity about the Wolfram page on Hofstadter's Male-Female Sequences: http://mathworld.wolfram.com/HofstadterMale-FemaleSequences.html Do you know what the two graphs represent? For sure ...
-2
votes
1answer
45 views

Show that shifting a sequence by $M$ positions does not change its covergence or limit. [on hold]

Suppose $(a_n)$ is a sequence, and $M$ is a fixed positive integer. We define a new sequence $(b_n)$ by $b_n = a_{M+n}$. (so the new sequence is the old one ‘shifted’ by $M$ terms.) Exercise 1. ...
0
votes
1answer
44 views

Proof of a series law

I'm stuck on the following exercise: "Let $\sum_{n=m}^\infty a_n$ be a series of real numbers, and let $k\geq 0$ be an integer. If one of the two series $\sum_{n=m}^\infty a_n$ and $\sum_{n=m+k}^\...
-5
votes
1answer
117 views

Telescoping function Revealed. [on hold]

Part B: I found Summation $$\sum_{n=0}^\infty\frac{x^n \ (-1)^n}{(y+1)^{n+1}} = \frac{1}{(x+y+1)}$$ However $x$ is related to $y$. $y \ge |x|$. You may derive the result and present it by ...
4
votes
3answers
163 views

Find the closed form for $\int_{0}^{\infty}\cos{x}\ln\left({1+e^{-x}\over 1-e^{-x}}\right)dx=\sum_{n=0}^{\infty}{1\over n^2+(n+1)^2}$

$$I=\int_{0}^{\infty}\cos{x}\ln\left({1+e^{-x}\over 1-e^{-x}}\right)dx=\sum_{n=0}^{\infty}{1\over n^2+(n+1)^2}\tag1$$ $$\ln\left({1+e^{-x}\over 1-e^{-x}}\right)=2\sum_{n=0}^{\infty}{e^{-(2n+1)x}\...
1
vote
1answer
16 views

How can I find the subsequential limit, limit sup, and limit inf of $s_n=n\tan\frac{n\pi}{3}$

$s_n=n\tan\frac{n\pi}{3}$ How can this sequence be decomposed to the the set of subsequences so that I can find the limit sup, and limit inf? I suppose I could just take $n$ and then $\tan\frac{n\pi}...
2
votes
0answers
72 views

Is there any other way to solve this question?

The sixth term of an arithmetic progression is $2$, and its common difference is greater than $1$. Show that the value of the common difference of the progression so that the product of the first, ...
17
votes
2answers
201 views
+300

Fibonorial of a fractional or complex argument

Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number$^{[1]}$$\!^{[2]}$$\!^{[3]}$. The Fibonacci numbers have a natural generalization to an analytic function of a complex argument: $$F(z)=\left(\phi^...