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

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7 views

Estimate the double series $\left|\sum_{n,m} b_n \bar{b}_m c_{n-m}\right|$.

I have some doubts on operations about the so-called "double series". For example, if we have $\{b_n\}$ sequence of complex numbers convergents in modulus. $\{c_{n-n}\}$ sequence of complex numbers ...
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1answer
38 views

Prove $\{s_n\}$ converges if $\{a_n = s_n + 2s_{n+1}\}$ converges.

Prove that $\{s_n\}$ is convergent if $\{a_n\}$ is convergent where $a_n = s_n + 2s_{n+1}$. This is an old (1950) Putnam question. Clearly $s_n + 2s_{n+1} \rightarrow L$. It looks obvious that $s_n ...
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1answer
24 views

compute very special limit in real number

Let the function $f:$ $\Bbb{R}\to\Bbb{R}$ such that $f(x)=\inf\{|x-me|:m\in\Bbb{Z}\}$ and consider sequence $\{f(n)\}$ then which of the following options is true? a) $\{f(n)\}$ is convergence b)the ...
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2answers
20 views

Convergence of series

Suppose $$\sum_{n=1}^\infty a_n$$converges with $$a_n>0 $$ ,show that $$\sum_{n=1}^\infty \frac{{a_n}^{1/2}}{n} $$ is convergent. Anyone can help me with this? Thanks!,prefer simple method!
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2answers
20 views

Sequence and series : Convergence

$\sum_{n=1}^\infty a_n,\sum_{n=1}^\infty b_n$ with $a_n, b_n >0 $ such that $\frac{a_{n+1}}{a_n} <= \frac{b_{n+1}}{b_n}, n>=\mathrm{some\space integer}$. Suppose $ \sum_{n=1}^\infty b_n$ ...
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1answer
42 views

Express $C_n = \cosh(0) + \cosh(1) + \cosh (2) + \dots + \cosh(n)$

Could someone give me some hint of how to do this question please. I've been stuck for more than $3$ hours on this question.
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3answers
56 views

$y_{2n}, y_{2n+1}$ and $y_{3n}$ all converge. What can we say about the sequence $ y_n$?

My friend and I are currently debating the following question: Let $y_n$ be a sequence in a metric space and assume that the subsequences $y_{2n}$, $y_{2n + 1}$, and $y_{3n}$ all converge. ...
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1answer
31 views

$\sum_{k=1}^\infty \frac {(-1)^{k+1}\cdot 1\cdot 4\cdot 7\cdots(3k-2)}{2^k\cdot k!}$

I'm working on: $$\sum_{k=1}^\infty \frac {(-1)^{k+1}\cdot 1\cdot 4\cdot 7\cdots(3k-2)}{2^k\cdot k!}$$ I've already shown that this series doesn't absolutely converge. I can't use Abel's test ...
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1answer
19 views

How to procede with alternating series

Given some alternating series, the first step is to check whether it's absolutely convergent. Say it's not. Then you use the alternating series test. That test tells you if the series is ...
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1answer
26 views

Define $(a_n)_{n\in \mathbb{N}}$ by $a_n:=\prod_{k=2}^n 1-\frac{1}{k^{1+c}}$. Does $\lim_{n\to\infty} a_n=0$ hold? [on hold]

Let $c\in \mathbb{R},c>0$ and define the sequence $(a_n)_{n\in \mathbb{N}}$ by $a_n:=\prod_{k=2}^n 1-\frac{1}{k^{1+c}}$. Does $\lim_{n\to\infty} a_n=0$ hold?
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2answers
37 views

General term of the sequence $2, 7, 16, 30, 50, 77, 112, 156$ [on hold]

What should be the general term of the sequence $2, 7, 16, 30, 50, 77, 112, 156$?
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1answer
30 views

What sequence of polynomials is equal to $2^n$ for integers $1$ to $k$?

I am trying to prove to someone that no matter how many terms you have of a sequence you can never be 100% sure of the underlying formula. Consider this sequence: $$2^n=1,2,4,8,16,...$$ But just given ...
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1answer
20 views

Sum of two divergent sequences with different number of partial limits

Suppose that $(a_n)$ is a sequence which has $1050$ partial limits, and $(b_n)$ is a sequence which has $2750$ partial limits. I'm asked to prove that $(a_n+b_n)$ diverges. So, in general the sum of ...
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1answer
33 views

Analyzing convergence of series with sine and cosine

Analyze the convergence of the following series: $$\displaystyle\sum\frac{\cos{n}}{\sqrt{n}+\cos{n}}$$ $$\displaystyle\sum\frac{\sin{n}}{\sqrt{n}+\cos{n}}$$ I tried to use the direct comparison test ...
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1answer
33 views

The proof of finding extreme points of the unit ball of $l^1$

Can someone show how to start the proof of finding extreme points of the unit ball of $l^1$? Thanks. Edit: How I've done so far is that Let $B$ be the closed unit ball of $l^1$ Consider any $x_n ...
2
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0answers
30 views

The series may converge, but what about the series / n?

Let $a_i$ be a positive sequence such that $a_i \to 0$. I know that the series $\sum_{i=1}^\infty a_i$ may be divergent. But what about the series divided by $n$; does the following go to 0? ...
0
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1answer
19 views

equivalency of the least upper bound property & convergence of every monotone and bounded sequence in $\mathbb{R}$

I'm aware how to prove convergence of every monotone and bounded sequence in $\mathbb{R}$ by using the completeness of $\mathbb{R}$ (using least upper bound property). But now I want to prove the ...
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2answers
33 views

Can we solve recurrence relations indexed by R?

Recurrence relations are equations with functions from $\mathbb N\rightarrow \mathbb N$. For example given $a:\mathbb N\rightarrow \mathbb N$ (write $a(n)$ as $a_n$) solving the recurrence relation ...
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0answers
20 views

Dirichlet integral using real Analysis

The teacher made this approach to solve the Dirichlet integral , $$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin ...
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2answers
115 views

Is there a closed formula for $1^1\cdot 2^2\cdot 3^3\cdot 4^4\cdot\ldots\cdot n^n$?

Calculate the product $$1^1\cdot 2^2\cdot 3^3\cdot 4^4\cdot 5^5 \cdot 6^6\cdot\ldots\cdot n^n$$ I have googled it alot but not found any solution. If there exists any formula to calculate product ...
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0answers
26 views

How can I ask a question in meta about posting some results of an analysis of certain multi-segment integer sequences? [on hold]

I have been doing some investigation using a computer program of multi-segment integer sequences. The segments are generated when you interrupt a Fibonacci-like sequence after a specified number of ...
2
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0answers
58 views

Partials sums of cosh(x) and sinh(x)

Ok, i asked this question yesterday but then hit a snag again. Using these identities $\sinh(x+1) - \sinh (x) = (-1+\cosh(1))\sinh(x) + \sinh(1)\cosh(x)$ $\cosh (x+1) - \cosh (x) = ...
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1answer
37 views

Closed form for Numbers in a Triangular Array

I have a particular triangular array $$ \begin{matrix} 1 & \\ 1 & 1 \\ 1 & 2 & 3\\ 1 & 3 & 9 & 15\\ 1 & 4 & 18 & 60 & 105\\ 1 & 5 & 30 & 150 ...
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2answers
45 views

Calculating $a_n$ in $\sum_{n=1}^\infty a_n \sin(\frac{n \pi}{2})=T_0$

I'm looking to solve the following when $T_0$ is a constant: $$\sum_{n=1}^\infty a_n \sin\left(\frac{n \pi}{2}\right)=T_0$$ If it matters this was reached from the following: ...
1
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1answer
31 views

Is there any upper bound of this sum?

$a_1,a_2,\ldots,a_n,k$ are all integers. Is there any upper bound of the following sum $$\sum_{a_1+a_2+\cdots+a_n=k\textrm{ and } a_1,a_2,\ldots,a_n\ge 0} \frac{1}{a_1!a_2!\cdots a_n!},$$ which is a ...
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6answers
189 views

How to Find the Function of a Given Power Series?

(Please see edit below; I originally asked how to find a power series expansion of a given function, but I now wanted to know how to do the reverse case.) Can someone please explain how to find the ...
2
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4answers
98 views

Write a function as $\sum _{n=0} ^{\infty} a_n x^n$

We have $f(x) = (x+ x^2 + x^3 + x^4 + x^5 + x^6)^4$. Now I want to write this as $\sum _{n=0} ^{\infty} a_n x^n$. What I got: $f(x) = (x+ x^2 + x^3 + x^4 + x^5 + x^6)^4 = x^4 (1+ x + x^2 + x^3 + ...
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4answers
189 views

Proving $\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}} < 3$ for $n\geq 1$ by induction

How can I prove by induction that $\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}} < 3$ for $n\geq 1$? My guess is that there must be another form to express the sum of nested square roots, but I don't know how ...
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2answers
32 views

Does the series $\sum_{i=1}^{\infty}\log(\sec(\frac1{n}))$ converge?

Does the series $\sum_{i=1}^{\infty}\log(\sec(\frac1{n}))$ converge? My try:As $n$ approaches zero, $\sec(\frac 1n)$ gets close to $\frac 1{1-0.5\frac 1{n^2}}=\frac{2n^2}{2n^2-1}=1+\frac ...
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2answers
54 views

Interesting $0, 1$ sequence of numbers,after $n>2, a_n$is composite.

Let us have a finite sequence with only $0$ and $1$ digit in our numbers(it can begin with $0$ too). $a_n$ is the number, which we get if we write our number $n$ times next to each other. Prove, that ...
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1answer
138 views

Isn't really a monotonic sequence?

First, I'd to say that I'm a beginner so may you answer easily plz. I'll expose you the problem: I was looking up on this page http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF and find ...
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2answers
32 views

Prove that $\sum_{n=0}^\infty \frac{1}{|\alpha_n-a|}\leq\sum_{n=0}^\infty \frac{C}{|\alpha_n|}$.

Let us consider the series of real numbers $$\sum_{n=0}^\infty \frac{1}{|\alpha_n|}$$ and we assume it convergent. What can we say on $$\sum_{n=0}^\infty \frac{1}{|\alpha_n-a|}$$ for each $a\in\mathbb ...
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2answers
29 views

Convergence of a sequence of integrals

I've tried expanding the hinted expression by using the definition from part (i) and choosing an X0 sufficiently large that |f(x)-l| < 1 but this doesn't appear to help very much at all. I've ...
2
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1answer
46 views

Random variable with infinite expectation

I was trying to find $Y$,a random variable (non-negative, may be $E(|Y|)=\infty$), such that $$\sum_{n=0}^{\infty} E\Bigl(\frac{|Y|}{n^2 +|Y|}\Bigr)=\infty$$ I tried with Cauchy distribution but could ...
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3answers
29 views

How to prove that those limits are equal?

let $\lim a_n = a$ and $\lim b_n = b$. We define two groups:- Group A:- Includes all the elements that imply $a_n>b_n$.All the elements in $a_n$ that are bigger than the the elements in $b_n$. ...
5
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1answer
35 views

A Limit of a Geometric Average

I have a problem calculating the following limit: $\lim\limits_{n \to \infty}{ (1-2/3)^{3/n}*(1-2/4)^{4/n}...(1-2/(n+2))^\frac{n+2}{n}}$ I thought this is a geometric average of the first n items of ...
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4answers
47 views

Maclaurin series of function $f(x)= x+\ln(9-x^2)^\frac{1}{3}$

Find the Macluarin series. I'm trying for hours to understand how should I solve it. Please explain it to me step by step. $f(x)=x+\ln {\sqrt[3]{9-x^2}}$
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2answers
24 views

How do I prove this statement about limits?

If $a_n$ is monotonic increasing and $b_n$ is a Bounded series and the limit of $a_n$ - $b_n$ is zero then prove that $b_n$ has a limit.I know that if I proved that $b_n$ is monotonic ...
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2answers
23 views

Norm on the space of sequences

Given the sequence spaces $\ell^p$ that are defined as: $$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$ for $\infty > p ≥ 1$, I'm trying to show that ...
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2answers
42 views

$\sum_{n=2}^\infty {(-2)^n \over n} $ How does this converge or diverge using the alternate series test?

$$\sum_{n=2}^\infty {(-2)^n \over n} $$ When I took the limit I got -2, I also tried using ratio and root test and got the same answer. The answer is supposed to be divergent I think but I thought if ...
4
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1answer
49 views

If $\limsup_n\sqrt[n]{a_n}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{(n+1)a_{n+1}}=?$

If $\limsup_n\sqrt[n]{|a_n|}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{|(n+1)a_{n+1}|}=?$ Can I separate the product of the second limit as ...
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1answer
44 views

Sum of hyperbolic functions, having problems expressing $\sinh(1)$

Using these identities $\sinh(x+1) - \sinh (x) = (-1+\cosh(1))\sinh(x) + \sinh(1)\cosh(x)$ $\cosh (x+1) - \cosh (x) = (-1+\cosh(1))\cosh(x) + \sinh(1)\sinh(x)$ Express the series $C = \cosh 0 + ...
5
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7answers
102 views

Show convergence/divergence for $\sum_{n=1}^{\infty} \frac{{(\ln n)}^{2}}{{n}^{2}} $

$$\sum_{n=1}^{\infty} \frac{{(\ln n)}^{2}}{{n}^{2}} $$ Anyone can give hint for this? Thank you!
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1answer
51 views

Limit comparison test

Since the series $\sum_{n=1}^\infty a_n$ is convergent, so $\lim_{n\to\infty} a_n=0.$ Consider $$\lim_{n\to\infty} {\left(\dfrac{{a_n}^{1/2}}{n}\right)\over{\left(\dfrac{1}{n^2}\right)}} $$ which is ...
2
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3answers
43 views

Finding the Limit of the Ratio of a Recursive Sequence's Terms

Let {$f_n$} be defined recursively as $f_1 = f_2 = f_3 = 1$ and $f_n = f_{n-1} + f_{n-3}$ for all $n \gt 3$. Also, define {$a_n$} as the ratio of the terms of {$f_n$}. That is, $a_n = ...
2
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2answers
46 views

How to prove convergence of the Arithmetic-Geometric Mean sequence?

Let $$a_0>b_0>0 $$and consider the infinite sequences $$\{a_n\}, \{b_n\}$$ where $$a_{n+1}=\frac{a_n+b_n}{2}$$ and $${b_{n+1}}={(a_nb_n)}^{1/2} $$ for $n\geq0$. Prove that the infinite ...
0
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0answers
19 views

Requesting help on understanding series [on hold]

Is the tangent of a positive convergent series still positive?
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0answers
50 views

a friend posted this on facebook i need help to figure it out [on hold]

She posted this fraction on facebook five days ago 148/13. And then yesterday she posted 145/15 and today she posted 145/18 she said the goal is a whole number what would that number be?
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0answers
26 views

Number of Unique Permutations of 3 digits (-1,0,1) given a length that match a sum

Say you have a vertical game board of n length (length being number of spaces). And you have a three sided die that has the options: go forward one, go back one, and stay. If you go below or above ...
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1answer
32 views

I have a feeling that these statements on my homework are true, but how would I prove it? [on hold]

If $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$ are both absolutely convergent series with all positive terms then $\sum_{n=1}^{\infty}a_n/b_n$ is absolutely convergent. If the power ...