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

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Degree of a Sequence of Squares Recurrence Relation

We are told that there exists a k-th order homogeneous linear recurrence relation $a_n=r_1a_{n-1}+...+r_ka_{n-k}$ which has distinct roots. We need to prove that for every $a_n$ that is satisfied by ...
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0answers
20 views

Hilbert's double series theorem $\sum_{n,m}\frac{a_n b_m}{m+n}$ and its generalizations.

Let $l_p$ be the space of all complex sequences $x=\{x_n\}$ equipped with the finite norm $\|x\|_p=\left(\sum_n |x_n|^p\right)^{1/p}$. Hilbert's double series theorem states that $$\sum_{n,m}\frac{a_n ...
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0answers
10 views

Limit of sequence of analytic functions

If $\Omega_1$ and $\Omega_2$ are two nonempty disjoint open subsets ${\bf C}$ and $\{f_n\}$ is a sequence of analytic functions from $\Omega_1 \to \Omega_2$ which converges pointwise to a function $f ...
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4answers
110 views

Does $\sum 3^{-\sqrt{n}}$ converge or diverge?

I need to find out whether this series converges or diverges: $$\sum_{n=1}^\infty \frac 1{3^{\sqrt{n}}}$$ The $n$th term, ratio, and root tests are inconclusive, Abel's test doesn't apply (or I ...
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1answer
22 views

Manipulating series to find the recursive formula

Ok so I am stuck. I need to get all the $n$'s to $=0$ but I can't reduce my series which has $n=2$ to $0$ because then I will have undone all my work in the first place to get all the $X^n$'s to the ...
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1answer
31 views

Help please with finding the equation and pattern of Taylor Series. (2 problems I have attempted down below).

I didn't want to ask twice so I combined both of my questions together. I have just started on Taylor Series, and I'm not very good at figuring out patterns. First Question Find Taylor Series for ...
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2answers
28 views

How can I find the radius and interval of convergece of $\sum_{n=0}^\infty {(x+5)^n} $, and for what value of x does the series converge?

$$\sum_{n=0}^\infty {(x+5)^n} $$ We talked about this briefly but I'm still pretty confused about how to start this problem.
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2answers
49 views

Find the smallest $N$ such that $\sum_{k=1}^N\frac{1}{p_k}>\pi$. (The $p_k$'s are the prime numbers.)

How to solve the following problem? Let $\{p_k\}_{k=1}^\infty$ be the set of primes (in increasing order). What is the smallest integer $N$ such that $$\sum_{k=1}^N \frac{1}{p_k}>\pi?$$ We ...
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1answer
32 views

Test $\sum\limits_{n=1}^{\infty} \frac{(-1)^n\ln(n)}{n}$ for convergence

$\sum\limits_{n=1}^{\infty} \frac{(-1)^n\ln(n)}{n}$ The first thing I think to do is the alternating series test, but $\frac{\ln(n)}{n}$ is not a monotonically decreasing sequence. For example, the ...
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28 views

What to do when Ramanujan summation diverges too?

While using Ramanujan summation to some kind of divergent series I got stuck: let's take the definition of this sum for the terms of a general function $f(x)$: $$\Re(x)=\int_n^xf(t)dt-\frac ...
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27 views

An inequality of the type $(\text{const.})\sum_{i=1}^\infty b_i^2\leq\sum_{i=1}^\infty \left( \sum_{j=1}^\infty a_{i,j} \right)^2$

Let us consider a sequence of real numbers. It is known that $$\sum_{i=1}^N a_i^2 \leq 4\sum_{i=1}^N \left( \sum_{j=1}^i a_j \right)^2\ \ \ (*)$$ I have a curiosity. If we have a double series: ...
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1answer
27 views

Proving a sequence is convergent and calculating its limit

In my assignment I have to solve the following question. I think I have an idea how to solve it, but I suspect there is a little thing in my solution which is wrong. If you can tell if my solution is ...
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2answers
44 views

How to show that there exists a sequence in $[0,1]$ such that the set of accumulation points of the sequence is $[0,1]$

This is related to homework but I am trying to find a special case first and see if I can generalize it. The problem is to construct some sequence $(x_n)$ in $[0,1]$ such that the accumulation points ...
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0answers
17 views

$\lim\sum_{k=0}^{\lfloor\delta n\rfloor} \frac{n^k}{k!}e^{-n}$ and Poisson distribution

Problem: Let $X_1,X_2\ldots$ be some independent random variables with Poisson distribution with parameter 1. Show that for every $\epsilon > 0$ sequence $S_n-(1-\epsilon)n$ converges to ...
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2answers
28 views

Convergence: infinite series

$\sum_{n=1}^\infty a_n,\sum_{n=1}^\infty b_n$ with $a_n, b_n >0 $ such that $\frac{a_{n+1}}{a_n} \leq \frac{b_{n+1}}{b_n}, n\geq\text{some integer}$. Suppose $ \sum_{n=1}^\infty b_n$ ...
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3answers
51 views

Is $\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$ always?

I wonder, whether it is always the case $$\sum_{k=-\infty}^{0}f(-k)=\sum_{k=0}^{\infty} f(k)$$ in regards of summation methods for divergent series?
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1answer
27 views

Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...
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1answer
42 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
29 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
29 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
26 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
73 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
64 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
35 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
24 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
38 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
23 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
36 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
34 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 ...
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0answers
31 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? ...
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1answer
20 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
35 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
24 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
121 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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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 ...
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1answer
113 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
39 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
55 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: ...
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1answer
33 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
193 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 ...
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4answers
100 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
193 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
35 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
55 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
171 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
35 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
30 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
47 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 ...