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

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0
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1answer
147 views

I have a question about integrating, and what to do about the constant. $\displaystyle\int\frac{1}{1-z}dz$

http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf On page 44, they conclude that $g'(z) = - \displaystyle\frac{\ln(1-z)}{z}$ by saying that it is just ...
0
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0answers
20 views

Randomness of a linear congruential genarator

I am working on a school project and it requires a simple pseudo-random number generator. I thought of using a linear congruential generator for this purpose. Here's a link ...
1
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2answers
41 views

For what sequences $a_n$ does the sequence $(1+\alpha a_n)^n$ converge?

We know $ (1+\alpha/n)^n \rightarrow e^{\alpha} $ when $n\rightarrow +\infty$. Suppose we are given a modified version of the problem: $$ \quad (1+\alpha\cdot a_n)^n \tag{1} $$ The question ...
1
vote
1answer
39 views

Find the supremum and infimum of $A = \left \{ \frac{(-1)^{n}n}{(-1)^{n + 1} n + 1}, n \in \mathbb{N}\right \}$

$$A = \left \{ \frac{(-1)^{n}n}{(-1)^{n + 1} n + 1}, n \in \mathbb{N}\right \}$$ So the supremum is $\frac{-1}{2}$, and the infimum would be $-1$, right? However the solutions say that the infimum = ...
0
votes
1answer
67 views

How to calculate the sum of an infinite series [duplicate]

How do you calculate the sum of an infinite series like $$ \sum_{n = 0}^\infty \frac{n}{2^\sqrt{n}}$$ //EDIT //Ignore I searched up how to find this with infinite geometric series solution which ...
1
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1answer
57 views

Behaviour of the sequence $u_n = \frac{\sqrt{n}}{4^n}\binom{2n}{n}$

Let $n\in \mathbb{N}^{*}$, let ${\displaystyle u_{n}={2n \choose n}\sqrt{n}\ 4^{-n}}$ Show that $(u_{n})_{n}$ is convergent and ${l.e^{-\frac{1}{8n}}<u_{n}<l}$ The original text i'm ...
0
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3answers
39 views

convergence of infinite series $\sum_{n=1}^\infty \frac{x^n}{(1+x)(1+x^2)(1+x^3)\cdot\cdot\cdot (1+ x^n)}$

I am reviewing for my final exam, and viewed this question: Decide whether the following infinite sum is convergent for all $x >1$: $$\sum_{n=1}^\infty ...
1
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2answers
33 views

convergence of the series $\sum_{k=0}^{\infty}E_k(x)$

Define $E_0(x)=x$, $E_1(x)=e^{E_0(x)}=e^x$, $E_2(x)=e^{E_1(x)}=e^{e^x}$, $\cdots$, $E_{n+1}(x)=e^{E_n(x)}$. For which values of $ x $ the series $\sum_{k=0}^{\infty}E_k(x)$ converges?
1
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1answer
40 views

Proving by defintion: $\lim_{n\to\infty}\frac {1-n^2}{2n^2+n+5}=-\frac 1 2$

Prove by definition that: $\displaystyle\lim_{n\to\infty}\frac {1-n^2}{2n^2+n+5}=-\frac 1 2$ Scratch work to find $N\in \mathbb R$: $|\frac {1-n^2}{2n^2+n+5}+\frac 1 2|=\frac ...
3
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4answers
57 views

A problem with proving using defintion that $\lim_{n\to\infty}\frac {n^2-1}{n^2+1}=1$

Prove using the definition that: $$\displaystyle\lim_{n\to\infty}\frac {n^2-1}{n^2+1}=1 $$ What I did: Let $\epsilon >0$, finding $N$: $\mid\frac {n^2-1}{n^2+1}-1\mid=\mid\frac ...
1
vote
1answer
138 views

What is $\sum_{n=1}^{\infty} \frac{n}{2^{\sqrt{n}}}$?

What is the value of $$\sum_{n=1}^{\infty} \frac{n}{2^{\sqrt{n}}}$$ It's clearly convergent but is it possible to calculate the sum?
0
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2answers
667 views

prove - if a tends to L then Ma tends to ML

Suppose that the sequence $(a_n)_{n=1}^{\infty}$ is such that $a_n \rightarrow L$ as $n\rightarrow\infty$. Prove that for any $M>0, M \in \mathbb{R}$, we have that $Ma_n \rightarrow ML$ as $n ...
2
votes
3answers
92 views

The limit of sequence tends to $0$

I am trying to show that if $0<x<1$, $$ \lim_{n\to \infty} {n^2 x^n (1-x)}=0 $$ I can't think of a clever way to show it.
1
vote
1answer
54 views

Proving completeness of subset in $\mathbb{R}^2$

I am struggling with this question about completeness of subsets of sequences. -Show if the following subset of $\mathbb{R}^2$ with standard metric is complete; if it is, prove; if not, find one ...
5
votes
2answers
91 views

Study the convergence of $\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$

Can you help me to study the convergence of the following series: $$\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$$ Thanks.
1
vote
1answer
36 views

Converging series

Suppose that $\sum_{n=0}^\infty a_{n}$ is a convergent series, with $a_{n}\gt0$ and suppose that $b_{n}\gt0$ is a bounded sequence. Then show that the series $\sum_{n=0}^\infty (a_{n}b_{n}$) is ...
3
votes
1answer
37 views

A property about tail equivalent random variables

Let $(X_n)$ and $(Y_n)$ be tail equivalent random variables i.e. $\sum_{i=1}^{\infty}\mathbb P(X_i\neq Y_i)<\infty$ Show that $\sum_{n=1}^{\infty}X_n$ and $\sum_{n=1}^{\infty}Y_n$ converge or ...
1
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2answers
38 views

two recursive sequences and the limit of their quotient

The sequences $\left \{ a_{n} \right \}$ and $\left \{ b_{n} \right \}$ are defined by the following recurrence relations: $a_{1}=b_{1}=1$ $a_{n+1}=a_{n}+2b_{n}$ $b_{n+1}=a_{n}+b_{n}$ What ...
0
votes
1answer
14 views

Differentiation equations and Series Expansion

When progressing through a worksheet for series I've stumbled across these form of problems which seem to boggle my mind. It does seem like it requires Maclaurins series however the approach seems to ...
1
vote
1answer
78 views

How to prove this Catalan number identity

Catalan number is $\displaystyle C_n= \frac{1}{n+1}\binom{2n}{n}$. How to prove that $$C_{2n-1} = \sum_{k=0}^{n-1}\left(\binom{2n-1}{n-k-1}-\binom{2n-1}{n-k-2}\right)^2$$ for $n\geq 1$. Thank you.
7
votes
4answers
132 views

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to?

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to? As far as I can tell, this has no closed-form solution (not saying much, I don't know much math), but a friend of mine swears he saw a ...
4
votes
1answer
59 views

Recurrent problem about polynomials

Given is a sequence of polynomials $P_n$, defined as follows: $P_0(x)=0, P_{n+1}(x) = P_n(x) + \frac{x-P_n^2(x)}{2}. $, n= 0,1,2,..., and x is real. Proving that for all non-negative integers n and ...
1
vote
2answers
452 views

Concentration of a drug after repeated injections

After injection of a dose $D$ of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as $D\exp^{-at}$ where $t$ represents time in hours and $a$ ...
1
vote
1answer
56 views

Can we find two convergente series $B=∑_{n=1}^{∞}b_{n}$ and $C=∑_{n=1}^{∞}c_{n}$ such that $A=BC$ [closed]

Let $A=∑_{n=1}^{∞}a_{n}$ be any convergente series. Can we find two convergente series $B=∑_{n=1}^{∞}b_{n}$ and $C=∑_{n=1}^{∞}c_{n}$ ($b_{n}$ and $c_{n}$ depends on $a_{n}$) such that $$A=BC$$
-3
votes
1answer
47 views

I don't know the answer [closed]

Let we have the sequence of real numbers $\left(\frac1{n^{1/n}} -1\right)$ clearly this sequence converges to zero , but I think it is not in any Lp space .
11
votes
2answers
204 views

How to prove this series $\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$ diverges

Question: Assume that $a_{n}>0,n\in N^{+}$, and that $$\sum_{n=1}^{\infty}a_{n}$$ is convergent. Show that $$\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$$ is divergent? My idea: since ...
0
votes
2answers
56 views

Two series with $a_n/b_n\to 1$ converges simultaneously?

For two series $\sum a_n$ and $\sum b_n$, if $a_n/b_n\to 1, (n\to\infty)$, can we assert that if $\sum a_n$ converges, then $\sum b_n$ converges; if $\sum a_n$ diverges, then $\sum b_n$ diverges. I ...
4
votes
0answers
68 views

Does $\sum_{n=0}^N{1\over\sqrt{n!}}$ have a closed form?

Just out of curiosity, does the sum $$\sum_{n=0}^N{1\over\sqrt{n!}}$$ have a closed form for $N<\infty$ or eventually $N\to\infty$ ? I cannot find it anywhere and it does not resemble any function ...
0
votes
1answer
29 views

Simple question about recursive sequence format regarding $a(n+2) = -4a(n+1) + 5a(n)$

Suppose there's a recursive sequence $a(n+2) = -4a(n+1) + 5a(n)$ How can i convert it into the form $a(n)$ because I am most comfortable solving questions in this form. I tried to find out but I'm ...
0
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1answer
37 views

Calculus II: Radius of Convergence

I have this math problem that states: In each part, write out the first four terms of the series, and then find the radius of convergence. $$(a) ...
-2
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3answers
134 views

Explanation of series for $\sin(x)$ and $\cos(x)$. [closed]

Can anyone explain me what is this equation telling us? I need to implement it in my computer program. I do not need a proof of these, but an explanation of notation used here. $$ \sin x = ...
5
votes
1answer
68 views

Sum involving zeros of Bessel function

I came across the following sum in my work involving the infinite sum of function of zeros of Bessel functions. $$ \displaystyle ...
0
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0answers
18 views

Orbit closure is uncountable, unless there is a periodic element.

Let $a = (a_i)_{i \in \mathbb N}$ be a sequnece over some finite alphabet $\Sigma$. We may define on the space $X = \Sigma^{\mathbb N}$ a shift operation by $(Sx)_i = x_{i+1}$. Let $A$ be the orbit ...
0
votes
1answer
30 views

Taylor polynomials of degree n

I have this math question that states: Find the Taylor polynomials of degree $n$ approximating $ln(1+x)$ for $x$ near $0$. The $n$'s are 5, 7, and 9. $f^{(5)}(0)=24$; I got the derivative to ...
0
votes
4answers
48 views

How to prove the sequence given by $a_{n+1}=s+a_n^2$ is monotonic increasing?

Let $s$ be $0\:\le \:s\le \:\frac{1}{4}$ and consider this sequence: $a_1\:=\:s$ $a_{n+1}\:=\:s\:+\:a_n^2$ I want to prove that is monotonic sequence, so I thought about induction or assume in ...
2
votes
1answer
2k views

About Banach Spaces And Absolute Convergence Of Series

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
6
votes
6answers
293 views

Study the convergence of $\sum_{n=1}^{\infty}\Bigl( \sqrt[n]{1+\frac{1}{n}}-1\Bigr)$

I need to study the convergence of $$ \sum_{n=1}^{\infty}\biggl( \sqrt[n]{1+\frac{1}{n}}-1\biggr). $$ Any help appreciated!! Thanks!
2
votes
1answer
61 views

Convergence of $ \sum_{k=1}^{\infty} \sqrt{a} \prod_{i=1}^{k}\frac{1}{1+i a}$ as $a \rightarrow 0$

Using numerical simulation, I can see that $$ v(a)=\sum_{k=1}^{\infty} \sqrt{a} \prod_{i=1}^{k}\frac{1}{1+i a} $$ converges to some value $1<v(a)<2$ as $a \rightarrow 0$. However, I couldn't ...
4
votes
2answers
49 views

Show that the series $\sum \frac{\sin \left(\frac{\left( 3-4n \right)\pi }{6}\right) }{2^{n}}$ converges?

Using the addition formula for the sine function I have managed to reduce this to a simpler form: $$\sum \frac{\cos \frac{2n\pi }{3}}{2^{n}}$$ It is obvious here that it passes the n-th term ...
2
votes
4answers
96 views

Does $\sum_{n=1}^{\infty}\frac{\cos\left(\frac{n\pi}{2}\right)}{\sqrt{n}}$ converge?

Does the following series converge? $$\sum_{n=1}^{\infty}\frac{\cos\left({\frac{n\pi}{2}}\right)}{\sqrt{n}}$$ The $\cos$ function: alternates between (-1) and 1 for every $n$ that is even. ...
2
votes
1answer
36 views

Does it true that $\lim_{n\to\infty}\frac{\sin\left(\frac{1}{u_{n}}\right)}{\frac{1}{u_{n}}}=1$

We know that $$\lim_{x→0}\frac{\sin x}{x}=1$$ Let $(u_{n})_{n≥1}$ be any positive increasing sequence satisfying $\displaystyle\lim_{n\to\infty}u_{n}=+∞$ Can we deduce that ...
5
votes
2answers
62 views

deriving the sum of $x^n/(n+2)^2$

I am writing a research paper and I have stumbled upon an issue. I have to evaluate $$\sum_{n=1}^{\infty} \frac{x^n}{(n+2)^2}$$ Here is what I did: $$ \sum_{n=1}^{\infty} x^{n-1} = ...
2
votes
2answers
43 views

Infinite sequence of $\sqrt{2^n}$ equals $i$?

So I am no mathematician, in fact I consider myself not very good at math at all, however I do enjoy it. Anyways, I was messing around when I remembered a numberphile video I watched a while back ...
0
votes
1answer
23 views

Time series closed form similar to harmonic series

I want to find the closed form for the following: $$ S(k, \alpha) = \sum_{t=T-k}^T \frac{\alpha^{-t}}{t} $$ when $\alpha \in (0,1)$. For harmonic series there is an easy way to upperbound: $$ H(k) ...
1
vote
4answers
75 views

How to sum $\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2}$?

Does anyone know the general strategy for summing a series of the form: $$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2},$$ where $a$ is a positive integer? Any hints or ideas would be great!
0
votes
0answers
7 views

Need an explanation for Variational Iteration Method.

I read paper written by Onur Kiymaz and Aysegul Cetinkaya on 'Variational Iteration Method for a Class of Nonlinear Differential Equations'. On the first pages they introduced the method and later ...
1
vote
4answers
64 views

Show that an increasing sequence diverges if and only if it is unbounded.

Show that an increasing sequence diverges if and only if it is unbounded. How should I go about proving this?
0
votes
1answer
26 views

Stuck on a difference equation which requires an A-level method

In the non-zero sequence $x[n-1]+x[n+1]=ax[n]$ and $x[n+4]=-x[n]$ i) Find possible values of $a$. ii) For what values of $b$ is $b^n$ a solution ($x[n]=b^n$)? I need to solve this using only ...
1
vote
2answers
48 views
-4
votes
1answer
39 views

True or False? If true provide a proof. If false provide a counterexample. [closed]

A) Every subset of R has a least upper bound. B) If a sequence is not monotonic then it diverges. C) Let f : A -> B and g : B -> C be functions. g o f is surjective if and only if f and g are both ...