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

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0
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0answers
6 views

Limit of given expression

Let $\sum a_k=s$. I want to show that $$\lim\limits_{x\to 1^-}(1-x)\sum\limits_{k=1}^{\infty}\frac{ka_kx^k}{1-x^k}=s$$ where $x\in(0,1)$. Thanks for your helps.
1
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1answer
27 views

Proving the closed form for an infinite sum (related to Chebyshev polynomials)

How do I prove the following identity? For $y\not= 0$, we have $$ \sum_{n=0}^{\infty} \dfrac{1}{2y}\left( (x+y)^{n+1}-(x-y)^{n+1}\right) = \dfrac{1}{(x+y-1)(x-y-1)}. $$ I am trying to find the ...
1
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0answers
34 views

How to calculate the $n$-th member of sequence $a_{n+1}=\sqrt{y+a_{n}}$ [on hold]

I was searching for function that following: $a_{x+1}=\sqrt{y+a_{x}}$ and $a_{0}=0$ and I found only for $y=2$ or $y=0$. For $y=2$: $f(x,2)=2cos(\frac{\pi}{2^{x+1}})$. For $y=0$: $f(x,0)=t^{2^{-x}}$ ...
2
votes
2answers
39 views

Converge series such that permuting the termes will change the limit.

I know that for a series that converge, if we permute the element of the sum, the series doesn't necessarily converge. For exemple $$\sum_{n=1}^\infty \frac{(-1)^n}{n}$$ converge but if we first sum ...
3
votes
4answers
31 views

Difference of consecutive pairs of sequence terms tends to $0$

This seems an elementary problem, but I don't know of any reference to it in the literature. Consider the sequence $(a_n)_{n=1}^\infty$ of real numbers. Suppose ...
0
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0answers
27 views

How to separate self-defining values from sigma?

$$\sum_{k=1}^{m} \sum_{j=1}^{n} a_kx_j^{b_k+b_i} = \sum_{j=1}^{n} y_jx_j^{b_i}$$ What I need to do is solve $a$ for every $i$ given ($i$ is between 1 and $m$), so their result won't be composed of ...
4
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3answers
34 views

finding $a_1$ in an arithmetic progression

Given an arithmetic progression such that: $$a_{n+1}=\frac{9n^2-21n+10}{a_n}$$ How can I find the value of $a_1$? I tried using $a_{n+1}=a_1+nd$ but I think it's a loop.. Thanks.
0
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4answers
87 views

Limit of $\{a_n\}$, where $a_{n+1} = \sqrt{2+a_n}$

I am struggling with this question: Let $\{a_n\}$ be defined recursively by $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Find $\lim\limits_{n\to\infty}a_n$. HINT: Let $L=\lim\limits_{n\to\infty}a_n$. ...
6
votes
1answer
81 views

Prove that $2AB$ is square [duplicate]

Let $$A= 1! \cdot 2! \cdot 3! \cdots 1002!$$ $$B= 1004!\cdot 1005! \cdots 2006!$$ Prove that $2AB$ is square. Help guys, I tried, I really did but I couldn't.
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2answers
28 views

Determine subsequence of sequence [on hold]

I know the formal definition of a subsequence, but can't figure out how to find them for some particular sequence. Could someone show some of the methods for finding them? Thanks for replies.
1
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3answers
67 views

probability and expected value

Hey I am not sure if I thinking correctly on this question? In a carnival, there is game which charges you $3$ dollars to play a game. You win $1$ dollar for every consecutive head you get and you you ...
6
votes
1answer
87 views

Convergence of $\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)$

I'm looking at some notes from my previous complex variables course and I need help verifying some things about the convergence of $$ \prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right) $$ on compact ...
1
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2answers
68 views

Approximating $\tan61^\circ$ using a Taylor polynomial centered at $\frac \pi 3$ : how to proceed?

Here's what I have so far... I wrote a general approximation of $f(x)=\tan(x)$ , which then simplified a bit to this: $$\tan \left(\frac{61π}{180}\right) + ...
0
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0answers
23 views

How to solve following sequence equation?

Suppose I have a sequence ${p^*_j}$, $j=0,...,m$, satisfying relations $p^*_j=p^*_{j-1}p_{j-1,j}+p^*_{j}p_{jj}+p^*_{j+1}p_{j+1,j}$, with \begin{equation} p_{jj}=\frac{2j(m-j)}{m^2}, \end{equation} ...
2
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0answers
27 views

Choosing a center for a Taylor polynomial of $f(x)=\tan(x)$ to best estimate $\tan(61°)$

So I've converted $61°$ to radians: $$\frac{61\pi}{180^\circ} $$ Do I just look for a "clean" $x$ value nearby to this? Such as $\dfrac{π}{3}$ ?
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0answers
32 views

How to manipulate the bound on the summation

$$ B_n^{f^2}(x) = \sum_{k=1}^n\sum_{j=0}^{n-k} 2^{k-j} {j+k \choose j} \frac{d^j}{df^j}[f^k] B_{n,j+k}^f(x) $$ I am looking to have the bounds switched, can someone show me exactly how this is done? ...
0
votes
0answers
77 views

sum and infinity

If you have the sums $ (1+2+..+n) + (1+2+3+..+n-1)+ (1+2+3+..+n-2)+(1+2+3+..+n-3)+...+(1+2+3)+(1+2)+1$for large enough $n$ $$\frac {n^3}{3!} \approx (1+2+..+n) + (1+2+3+..+n-1)+ ...
4
votes
1answer
55 views

Prove a sequence converges using sub-sequences

Let there be a sequence $a_n$ The following sub-sequences converge: $a_{n^3},a^3_{2n+3}-a^3_{2n+4},a^2_{2n+3}-a^2_{2n+4},a_{2n+15}$ Prove: $a_n$ converges I think it has something to ...
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0answers
25 views

An integral involving the Gamma function

By utilizing the results of two previously asked questions, Series involving Laguerre polynomials and Integral of binomial coefficients, what is a resulting value of the integral \begin{align} ...
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0answers
14 views

limiting distribution of a function of joint normals

Let $Z_n=(X_{1,n},X_{2,n})\sim N(\mu,\Sigma_n)$ where $\mu=(0,0)'$ and $$\Sigma_n=\begin{bmatrix}a^2+\frac1n & ab \\ab & b^2+\frac1n\end{bmatrix}$$ Then where does ...
4
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0answers
46 views

A sequence avoiding 3-term power progressions

Rankin1 studied sequences of integers that avoid 3-term geometric progressions, $(a, a c, a c^2)$, e.g., $$\{1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, \ldots \} \;$$ So, $18$ is excluded ...
3
votes
8answers
2k views

After switching a lamp on and off infinitely many times in one minute, is it on or off? [duplicate]

So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another ...
0
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2answers
44 views

Determine the digit in a consecutive sequence of numbers

All positive integers are written in order, one after another $$1234567891011121314151617...$$ Which digits appears in the 206 787th position?
6
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3answers
66 views

Convergent $x_n,y_n$ and $x_n^{y_n}$ diverges

Let $x_n, y_n > 0$. I need an example of convergent sequences $x_n,y_n$ such that $x_n^{y_n}$ diverges. Could you help me?
4
votes
1answer
43 views

$f_n$ converges uniformly to f but $f^2_n$ fails to converge uniformly to $f^2$

Consider functional sequence $f_{n}$ which is differentiable on $\left(a,b\right)$. Find an example of $f_n$ converging uniformly to f on $\left(a,b\right)$ such that $f^2_n$ fail to converge ...
1
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1answer
25 views

What is this 2-D array of numbers called?

Define $c(n,r)$ ($n\in\Bbb N;r\in\Bbb Z$) by setting $c(0,-1)=-1$, $c(0,0)=1$, and $c(0,r)=0$ otherwise, with all further $c(n, r)$ given recursively by $$c(n+1,r)=rc(n,r-1)-(r+1)c(n,r)$$(rather in ...
1
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3answers
41 views

Problem about $\sum_{n=1}^{\infty}n\exp\left(-x\sqrt{n}\right)$

Consider $$\sum_{n=1}^{\infty}n\exp\left(-x\sqrt{n}\right)$$ A. sum of the series is bounded on its set of convergence B. sum of the series is continuous on its set of convergence The correct ...
1
vote
1answer
19 views

Uniform convergence on singleton

First, recall the definition of uniform convergence: Consider functions $f_{n}:A\rightarrow\mathbb{R}$. The sequence of functions $f_{n}$ converges uniformly on set A to limit function f if ...
2
votes
1answer
54 views

Boundedness of the function

Let $x\in(0,1)$ and $S_{n-1}=\sum\limits_{k=0}^{n-1}x^k$. Then define $f$ as the following :$$f(x)=\sum_{n=1}^{\infty}\left(\frac{nx^n}{S_{n-1}}-1\right)$$ I need to show that ...
4
votes
2answers
68 views

$\sum_{n=1}^{\infty} \frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)$ diverges [on hold]

How can I prove the divergence of the series $$\sum_{n=1}^{\infty} \left(\frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)\right) $$ if $f:\mathbb{N} \rightarrow \mathbb{N}$ is injective? $ $
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2answers
28 views

Show $\lim_{m \rightarrow \infty} \frac{1}{m^2+1} = 0$ using the $\varepsilon$-$N$ definition of convergence

Show that $$\lim_{m \rightarrow \infty} \frac{1}{m^2+1} = 0$$ using the $\varepsilon$-$N$ definition of convergence.
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0answers
34 views

Interpretations of divergent series

I have already bumped into this question and found it not quite useful for my own purposes. The usual way the sum of the infinite series is taught and defined in 1st year undergrad course is that its ...
0
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2answers
66 views

Summation of special series

Does anybody know how to evaluate $$\sum_{i=2}^n(i^2)\cdot{i\choose2}$$ How about the general case of $(i^k)*{i\choose2}$? A nice formula would be great!
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2answers
31 views

Summation of “products”

Ok, I am pretty sure I won't get an answer because the question is somewhat hard, and I have done research to no prevail but how can I find the summation of ab if I know the summation of a and the ...
2
votes
1answer
43 views

Prove that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$

Suppose $\{a_i\}_1^{\infty} \subset (0,1)$ a) $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$ b) Given $\beta \in (0,1)$, exhibit a sequence $\{a_i\}$ such that ...
8
votes
0answers
131 views

What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves?

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a ...
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votes
4answers
53 views

How do I determine $r$ in this geometric series $a+ar+ar^2+\cdots$? [on hold]

The geometric series $a+ar+ar^2+\cdots$ has a sum of $12$, and the terms involving odd powers of $r$ have a sum of 5. What is $r$? Thank you for any help .
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2answers
60 views

Evaluate the sum below [on hold]

Evaluate the following sum $$1*1!+2*2!+3*3!+....+1000*1000!$$ any help guys?
5
votes
5answers
186 views

Binomial coefficients in Geometric summation

Guys please help me find the sum given below. $$\sum_{k=j}^i\binom{i}{k}\binom{k}{j}\cdot 2^{k-j}$$ (NOTE):The two coefficients are multiplied by 2 power (k-j) I am using the formula: ...
1
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2answers
51 views

Show that the sequence $\left\{\frac{2n}{2n-1}\right\}$ is monotone by using $a_{n+1} - a_{n}$

Note: I am looking at the sequence itself, not the sequence of partial sums. Here's my attempt... Setting up: $$\left\{\frac{2(n+1)}{2(n+1)-1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$ ...
3
votes
5answers
608 views

Find the value of this series

what is the value of this series $$\sum_{n=1}^\infty \frac{n^2}{2^n} = \frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{25}{32}+\cdots$$ I really tried, but I couldn't, help guys?
1
vote
3answers
62 views

About the sum of $\sum_{n=1}^{\infty} \frac 1 {n(n+1)}$ [on hold]

Find the sum of $\displaystyle \sum_{n=1}^{\infty} \frac 1 {n(n+1)}$ So I can see that it's a telescopic sum: $\displaystyle \sum_{n=1}^{\infty} \frac 1 {n}-\frac 1 {n+1}$, but since the sum ...
1
vote
1answer
16 views

Recursive Equation : $X_t=-\sum_{j=1}^{\infty}\phi^{-j}W_{t+j}$

$X_t=\phi X_{t-1}+W_t$ $\Rightarrow X_{t-1}=\frac{1}{\phi} X_{t}-\frac{1}{\phi}W_t$ Where , $|\phi|>1$ . But how does the following recursion relation occur : ...
1
vote
4answers
67 views

Monotonicity and convergence of the sequence $a_n=\sum_{k=1}^{n}\frac{1}{k+n}$

Let we have the following sequence $(a_n)$ such that $$a_n=\sum_{k=1}^n\frac{1}{n+k}$$ How can I prove that $(a_n)$ is increasing bounded sequence, then prove it is convergent and find its limit?
1
vote
1answer
41 views

Prove that:$\sum_{i=1}^n\frac{1}{x_i}-\sum_{i<j}\frac{1}{x_i+x_j}+\sum_{i<j<k}\frac{1}{x_i+x_j+x_k}-\cdots+(-1)^{n-1}\frac{1}{x_1+\ldots+x_n}>0.$?

Is there someone show me how do I prove this , I guess this inequality hold only if $x=0$ . Let $x_1,x_2,\ldots,x_n>0$. Prove that ...
4
votes
2answers
211 views

Recursive sequence of square roots of previous elements

Bruckner, Bruckner, Thompson - Elementary Real Analysis $a_1 = 1$ and $a_{n+1} = \sqrt{a_1+ a_2 + .. + a_n}$ Show that $$\lim_{n \to \infty}\ \frac{a_n}{n} = \frac12$$ I cannot untangle the square ...
6
votes
2answers
93 views

$ \lim_{n\to+\infty} \frac{1\times 3\times \ldots \times (2n+1)}{2\times 4\times \ldots\times 2n}\times\frac{1}{\sqrt{n}}$

Knowing that : $$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$ $$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$$ ...
0
votes
1answer
24 views

proving convergence of this sequence and calculating limit

So i have this sequence: $a_{n}=1+ \frac{1}{3}\cos1 +\frac{1}{3^2}\cos2+ ... +\frac{1}{3^n}\cos(n) $ I have to prove it is convergent, and then calculate the limit. I'm not totally sure how to find ...
2
votes
1answer
63 views

How to solve this seemingly easy problem?

In short, I need to prove that: $\sin 2nx\not\to-\sin 2x\quad x\ne\frac{k\pi}2,k\in\Bbb Z,n\in\Bbb N\quad \text{as}\quad n\to\infty$ The biggest trouble is that I know little about $x$, not even ...
0
votes
6answers
325 views

calculating 2 sums of series

So I have these two series given. 1: $\displaystyle\sum_{n=1}^{\infty}\frac{\sin{(2n)}}{n(n+1)} $ And I have to show that this sum is $\leq$ 1. 2: ...