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

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7 views

Proof of Harmonic-Geometric Mean

Let $a_1$ and $b_1$ be any two positive numbers. Let $\alpha_{n+1} = \frac{2 \alpha_n \beta_n}{\alpha_n + \beta_n}$ and $\beta_{n+1} = \sqrt{\alpha_n \beta_n}$. Show that both sequences converge and ...
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0answers
5 views

Approximating ArcCos(x) without Radicals

Take $$f(x)=2x\arccos\left(\frac{x^2+d^2-1}{2xd}\right)$$ and try and find $$ I(x)=\int_{d-1}^{3}dx f(x) \sqrt{\left(\frac{x-1}{x}\right)}\left(3-x\right)^3 $$ You'll find the result is messy (see ...
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1answer
17 views

Determine the region of convergence of series of complex functions

I have this problem. Find the region of convergence of the following series of complex functions $$ \sum_{n=1}^\infty \frac{2^n}{z^{2n}+1} $$ The progress I have made so far is that when n goes to ...
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0answers
12 views

Show that if $a_{i -1} + a_{i}$ is a maximum for all $2 \le i \le n$, then $\sum_{i=1}^{n} a_{i}$ is a maximum? [on hold]

In particular, I would like to show that if N_{i} is such that for given values of $N_{i+1}$ and $N_{i-1}$, $\ln\left(N_{i-1}\right) + \ln\left(N_{i}\right)$ is a maximum for each $2 \le i \le n$, ...
1
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0answers
37 views

BMO Round 2 question

I need help with this BMO question: The first term $x_1$ of a sequence is $2014$. Each subsequent term of the sequence is defined in terms of the previous term. The iterative formula is ...
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2answers
37 views

Help me understand how to take derivative of the PDF of X~binom(n,p) with respect to p.

This is the solution I was given. My questions: Why is it summed from k=1 to x. Shouldn't it be from k=1 to n? (If not, why not?) What is happening to the first term from line 1 to line 2? When we ...
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1answer
16 views

Is there a systematic way of resolving sequences of dots in figures?

http://bibliotecadigital.ilce.edu.mx/sites/telesecundaria/tsm01g01v01/u02t04s01.html I wonder if there is a systematic way to get the formula of a sequence of dots in figures, to resolve it faster, ...
0
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1answer
20 views

Prove that a sequence can be enumerated using Catalan numbers

This problem is taken from R.P. Stanley’s Enumerative Combinatorics. Give bijective arguments to show that sequences of $n$ $1$'s and $n$ $-1$'s in which the sum of the first $i$ terms is ...
1
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1answer
25 views

Confused about this definition of limit superior.

The definition I am given is as follows: Let $(x_n)$ be a real valued sequence. For each positive integer $n$, let $s_n:=\sup\{x_m:m\geq n\}$. If $(s_n)$ converges, we denote its limit by ...
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0answers
16 views

How to show that $a_n + a_{n+r} + a_{n+2r} = 0$ for all $n ≥ 0$ if and only if $3r$ is divisible by $p^k − 1$?

Let ${a_n}$ $(n ≥ 0)$ be an m-sequence in $Z_p$ of period ${p^k − 1}$ where p is a prime and $k ≥ 1$. Suppose that $r$ is a natural number with $0 < r < p^k − 1$. Show that $a_n + a_{n+r} + ...
2
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1answer
17 views

Monotone Convergence Theorem (for real sequences) equivalent to the Least Upper Bound Property?

Some days ago I have asked this question to which André Nicolas gave a link to this paper which contained a proof of the Least Upper Bound Axiom from Monotone Convergence Theorem via Archimedian ...
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0answers
129 views

What general function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3…?

Look at this sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3... It is defined as follows: $$f(n)=\begin{cases} 3 &\text{if $n \bmod 7=6,0$}\\ 2 ...
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2answers
25 views

radius of convergence $\sum a_n z^n$

Find radius of convergence of power series $\sum a_n z^n$, where $ a_n $=number of divisors of $ n^{50}$. I thought of applying root test, but the a_n looks little trickyy.
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1answer
76 views

How does this person solve the Putnam problem?

Consider this: 2003 A1 Putnam Solution. I am only looking at A1 for Putnam 2003. The problem is here: Problem A1 2003 I would like to proceed step-by-step: I understand $ka_1 = a_1 + a_1 + ... ...
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1answer
28 views

How can I perform Partial Fractions Decomposition on a Telescoping Series involving Exponentials?

Given: $\sum_{k = 1}^\infty\dfrac{6^k}{(3^k - 2^k)(3^{k+1} - 2^{k+1})}$ The Partial Fractions Decomposition is: $\sum_{k = 1}^\infty(\dfrac{3^k}{3^k - 2^k} - \dfrac{3^{k+1}}{3^{k+1} - 2^{k+1}})$ ...
5
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2answers
65 views

Limit $I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$

Im a new participant in this mathematical forum, so this is one of that i couldn't solve it. $$I=\lim_{n \to \infty } \sqrt[n]{\int_0 ^1 x^{\frac{n(n+1)}{2}}(1-x)(1-x^2)\cdots(1-x^n)d x}$$ I've ...
1
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0answers
24 views

partial sum of convergent series

Let series $\sum^\infty a_n$ is convergent but not absolutly convergent. And $\sum^\infty a_n $=0. Denote $ s_k $ the partial sum $\sum_{n=1}^k a_n $, k=1,2.... then which of following ARE true. 1.$ ...
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1answer
25 views

Show that ${\{s_n}\}$ is bounded if $s_n = b_1r + b_2r^2 + … + b_nr^n$ and $0 < r < 1$.

Question: Let ${\{b_n}\}$ be a bounded sequence of nonnegative numbers and r be any number such that $0 \leq r < 1$. Define $s_n = b_1r + b_2r^2 + ... + b_nr^n$ for every index $n$. Use The ...
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4answers
94 views

What function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3… [on hold]

What function can I use to represent the next sequence: 2,2,2,2,2,3,3,2,2,2,2,2,3,3,2,2,2,2,2,3,3... I am looking for general solution.
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0answers
21 views

Finding the limit of the series

Find the limit of the sequence , can somebody help ,I'm struck.
1
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1answer
15 views

Series Expanding a Function: Complex Answer?

I have $$ f(x)=2\arccos\left(\frac{x}{2}\right)-x\sqrt{1-\frac{x^2}{4}} $$ and my friend says that the series of this function about $x=2$ (truncated to the first term) given $x\leq2$ is ...
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2answers
52 views

Prove that $\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$ is divergent

In this question I've asked to decide the convergency of the series $\sum_{p \in \mathcal P} \frac1{p\ln p}$. Now I ask you to show that the series $$\sum_{p \in \mathcal P} \frac1{p\ln \ln p}$$ ...
0
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0answers
21 views

Continued fraction approximation

Let $\theta\in\Bbb{R}_{\gt0}$. A) Prove that the convergents for the continued fraction expansion of $\theta$ give us better and better rational approximations to $\theta$. B) Suppose $\theta\notin ...
7
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2answers
137 views

how to compute this limit

compute $I=\lim\limits_{n\to+\infty}\sqrt[n]{\int\limits_0^1x^{n+1}(1-x)\cdots(1-x^n)dx}$ attempt: I tried to evaluate the integral $$\begin{align} ...
2
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2answers
38 views

Generating function for Pell numbers

Problem: The Pell numbers $p_n$ are defined by the recurrence relation \begin{align*} p_{n+1} = 2p_n + p_{n-1} \end{align*} for $n \geq 1$. The initial conditions are $p_0 = 0$ and $p_1 = 1$. a) ...
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0answers
21 views

Evaluate the limit of series [on hold]

Finding the limit of series Sigma [r^1/8(n^x-1/x +r^x-1/n)]/n^x+1 as n tends to infinity , where r can take values of natural numbers and x is a constant.
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2answers
15 views

Convergence of a function involving a characteristic function of a decreasing interval

Let $f_k: \mathbb{R} \rightarrow \mathbb{R} , f_k(x) = \frac{1}{\sqrt{x}}\chi_{\left[\frac{1}{2^{k+1}},\frac{1}{2^k}\right]}(x)$ For $k \rightarrow \infty $ the interval ...
2
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2answers
31 views

Uniform Convergence of Series $\sum_{n=1}^{\infty}\frac{\sin(x)^n}{n}$

I'm trying to show uniform convergence of a series of complex numbers, but I'm having trouble. The series is as follows: $$\sum_{n=1}^{\infty}\frac{\sin(x)^n}{n} \rm{~~~~~~for}~~~0<x<\pi/2$$ I ...
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1answer
25 views

Proving that something diverges to infinity.

So I'm trying to prove that the sum of 1/(2k+1) diverges to infinity. I thought about doing a comparison test with the harmonic series 1/k and multiplying the harmonic series by (1/3) so it is (1/3k). ...
1
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0answers
35 views

Continuity on $[a,b]$ implies uniform continuity on $[a,b]$

I don't understand the step underlined in green. I understand that for any $n$ , $|f(x_n)-f(y_n)|\geq \varepsilon$ where $x_n, y_n$ satisfy the conditions given regarding a function being not ...
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0answers
19 views

Is this a Cauchy sequence?

Let $Y$ and $Z$ be Banach spaces. Define $|u|_X := |Au|_Y + |Bu|_Z$ where $A$ and $B$ are linear maps. Suppose I have a sequence $(u_n)$ such that $|Au_n|_Y \to 0$ and $|Bu_n - Bu_m|_Z \to 0$. Does ...
2
votes
1answer
32 views

Prove that if ${\{{a_n}^2}\}$ converges (${\{a_n}\}$ is monotone), thus ${\{a_n}\}$ converges and to what?

From Fitzpatrick's Advanced Calculus book: "Suppose that the sequence ${\{a_n}\}$ is monotone, i.e., either monotonically increasing or decreasing. Prove that ${\{a_n}\}$ converges if and only if ...
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2answers
22 views

Convergent sequence as series, maximum of sequence as limit

I'm currently studying for my math exams. I came across two exercises about sequences and series for which I have no clue. So any hints would be appreciated. First problem: $(a_n)_{n\in\mathbb{N}}$ ...
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votes
3answers
30 views

Find range of $p$ such that the series converges

let $a_n$ be a sequence of real numbers such that the series $\sum |a_n|^2$ is convergent.Find range of $p$ such that the series $\sum |a_n|^p$ is convergent. My try: To show the series it is ...
1
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2answers
48 views

convergence of $\sum_{n=1}^\infty\frac{1}{n} [1+\frac{1}{\sqrt{2}}+…+\frac{1}{\sqrt{n}}]$ [duplicate]

Let $t_n= \frac{1}{n} [1+\frac{1}{\sqrt{2}}+.....+\frac{1}{\sqrt{n}}]$, n=1,2... Then I am asked whether series$ \sum t_n $ converge or diverge. Also whether sequence $ t_n $ converge to zero or not. ...
4
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1answer
51 views

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing?

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing? In process of solving this problem, I faced to the problem of proving that $A::$: ...
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0answers
14 views

Explicit formula of a sequence [on hold]

Do you know the explicit formula for this recursively defined sequence? $(a_{n+1})=(a_n)*2+1$
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2answers
701 views

Is the product of two monotone sequences monotone?

Question: The product of monotone sequences is monotone, T or F? Uncompleted Solution: There are four cases from considering each of two monotone sequences, increasing or decreasing. CASE I: Suppose ...
3
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1answer
49 views

Is $\lim na_n \to 0$ sufficient for $\sum a_n$ to converge?

Is $\lim na_n \to 0$ sufficient for $\sum a_n$ to converge? Or additional criteria is required? E.g. $a_n$ needs to be positive? Is naïve comparison with $\frac {1}{n^p}$ series justifies that ? Or is ...
0
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1answer
16 views

Find the ratio and interval of convergence for $\sum_{n=1}^{\infty} \frac{n!x^n}{1\cdot 3\cdot 5\cdots (2n-1)}$

I believe this would diverge for $x\neq 0$. After using the ratio test I obtain (x)(n+1)(sum from 1 to n of (2n-1)/(2n+1)). Taking the limit as n goes to infinity the second term blows up and the ...
3
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1answer
71 views

Prove the sequence determined by $a_{n+1}={a_n\over \sin a_n}$ is convergent, and found its limit.

Let $\{a_n\}$ be a sequence defined by $0<a_1<{\pi \over 2}$, $a_{n+1}={a_n\over \sin a_n}$. $Attempt:$ $a_1>0$ and $\sin a_1>0$ and therefore the sequence begins positive and remains ...
5
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1answer
58 views

Evaluating $\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k} $

Question: How to compute $$ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k}? $$ Here is what I have tried so far: Define $s_n=\sum\limits_{k=1}^n \frac{1}{k 2^k}$ for every index $n$, ...
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2answers
18 views

Find closed form for a sequence using the Fibonacci and Lucas number sequences. [on hold]

Define $A_n$ as follows: $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that ...
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3answers
49 views

Easy Analysis question [on hold]

Prove $\{\sqrt{n+1}-\sqrt{n}\}$, $n ≥ 0$, is monotone, using just algebra
2
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3answers
29 views

Explicit (or recursive) formula of a sequence

Is there an explicit or recursive formula for this sequence starting from n=1: 1, -1 , -1 , 1 , 1 , -1 , -1 , 1 , 1 , -1 , -1 , 1 , 1 , -1 , -1 , 1 , 1 , ...
0
votes
2answers
41 views

proving a limit by definition

given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a>1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition. proof: given $a_n=a^{\frac{1}{n}}$ for ...
0
votes
1answer
34 views

Convergent series proof!

Let $S$ be a non-empty subset of $\Bbb{R}$ that is bounded above. Show that there exists a sequence $(a_n)_{n\in \Bbb{N}}$ contained in $S$ (that is, $a_n \in S$ for all $n \in \Bbb{N}$) which is ...
0
votes
2answers
25 views

Determine if the series diverge or converge?

So I was wondering what is the best TEST(divergent test, alternating series,power series,ratio test or root test) that we can use for the following series :
1
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2answers
48 views

Infinite Sum with differential operator

How would one suggest to calculate the following sum? $\sum^{∞}_{n=1}\partial_{x}^{2n}(\frac{\pi}{2x}Erf[\frac{cx}{2}])=?$ where c is just a constant. cheers.
0
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0answers
13 views

need help solving this series [on hold]

i'm finding it difficult finding if this series converges or diverges. any help is appreciated. $\sum _{n=0}^{\infty }\left(3^{2+n}2^{1-3n}\right)$