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

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0
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1answer
40 views

Limit of recurrence relation $x_k= f(x_{k-1})$ for an increasing function $f:[0,1]\to[0,1]$

Is there a standard result out there that gives the following? It looks graphically like it must be true, but I'd like to appeal to a known result if possible. Let $f:[0,1]\to[0,1]$ be a continuous, ...
1
vote
3answers
57 views

How to calculate the integral of $\sum_{n=1}^\infty (1/r)^{n+1} r^2$?

How to calculate this integral? $$\int\limits_0^1 {\sum\limits_{n = 1}^\infty {\left( {\frac{1}{r}} \right)} } ^{n + 1} r^2 dx$$ Here $r$ is a real number
2
votes
3answers
97 views

Double summation index problem

I often meet the following situation: $$\sum\limits_{n=0} ^\infty \sum\limits_{k=0} ^n f(k)g(n-k)=\sum\limits_{p=0} ^\infty \sum\limits_{q=0}^\infty f(p)g(q)$$ While intuitively this is very clear ...
0
votes
1answer
29 views

Power of matrix expansion

We know about the expansion $(a+b)^n\tag 1$,for scalar variables. What will be the equivalent when we want to find $(A+B)^n \tag 2 $, when A and B are square matrices? Can we treat it as same as ...
3
votes
1answer
130 views

Finding the limit $\lim_{n\rightarrow \infty}\left(\sqrt{n^2+n+1}-\big\lfloor \sqrt{n^2+n+1} \big\rfloor \right)$

I need to find the value of the limit $$ \displaystyle \lim_{n\rightarrow \infty}\left(\sqrt{n^2+n+1}-\big\lfloor \sqrt{n^2+n+1} \big\rfloor \right),$$ where $n\in \mathbb{N}$. My attempt. When ...
6
votes
3answers
560 views

How do you calculate this sum?

How to find the value of $S(\infty)$, where $S(n)$ is the following $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
1
vote
0answers
23 views

'Deriving' the Laplace Transform from the z Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
1
vote
5answers
96 views

Existence of limit for a given sequence: $x_{n+1} \le x_n + 1/{n^2}$

Let $x_n$ be a sequence in $\mathbb{R}$ such that $$x_{n+1} \le x_n + \frac{1}{n^2}$$ Prove that $\lim x_n$ exists (it can be a real number or infinite). I've tried to prove it using the ...
0
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1answer
55 views

Calculation of a series

Calculate the series $$\sum^{\infty}_{n=1}\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}$$ I tried the root test but couldn't figure out, any ideas?
0
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1answer
31 views

$\Delta^kn^\alpha$ converges monotonically to zero when $\alpha<k$

I am trying to prove that the $k$-th finite difference of the series $n^\alpha$ converges to zero monotonically as $n\to\infty$ when $\alpha<k$. The differential analogue is ...
0
votes
4answers
70 views

Can this limit be solved in this manner?

$\lim _{n\to \infty }\left(\left(1+\frac{1}{1}\right)\cdot \left(1+\frac{1}{2}\right)^2\cdot ...\cdot \left(1+\frac{1}{n}\right)^n\right)^{\frac{1}{n}}$ Hi. I'm trying to solve this limit without ...
0
votes
3answers
46 views

Question on Sequences and limits

If sequence {$a_n$} satisfies $\displaystyle \lim_{n \to \infty} (2n-1)a_n=40$, what is the value of $\displaystyle \lim_{n \to \infty}na_n$ ? Any hints ?
1
vote
1answer
183 views

How to find a series for comparison with $\sum 1/\sqrt{n(n+1)}$?

The series $$\sum_{n=1}^{+\infty} \frac{1}{\sqrt{n(n+1)}}$$ I have tried ratio and integral both lead me to inconclusive, so probably it's by comparisson but I can't find What to compare.
3
votes
1answer
81 views

How do I use Euler's result to find the sum of a series?

So I am given: $$ \zeta(4) = \sum_{n=1}^\infty {1\over n^4}={\pi^4 \over 90} $$ I need to use it to find the sum of the following series using the above information. $$ \sum_{k=1}^\infty ...
2
votes
1answer
68 views

General term of sequence $a_0=2$ and $a_{n+1}=2a_n^2-1$

Is it possible to find general term of this sequence? $a_0=2$ and $a_{n+1}=2a_n^2-1$
8
votes
6answers
214 views

If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_n $, prove that $\sum \frac{a_n}{b_n}$ diverges

Let $\displaystyle \sum a_n$ be convergent series of positive terms and set $\displaystyle b_n=\sum_{k=n}^{\infty}a_n$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges. I could ...
6
votes
0answers
62 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
12
votes
6answers
2k views

Which number remains alive?

There are $100$ people standing in a circle numbered from $1$ to $100$. The first person is having a sword and kills the the person standing next to him clockwise i.e $1$ kills $2$ and so on. Which is ...
2
votes
2answers
72 views

How should I go about solving the following limit?

How do I solve the following limit? $$\lim _{x\to \:1} \frac{\left(1-x^{1/2}\right)\left(1-x^{1/3}\right)\cdots \left(1-x^{1/n}\right)}{(1-x)^{n-1}}$$ The solution should be 0, but the source isn't ...
0
votes
0answers
23 views

analyticical solution of a Recurrence serie

I have the following recurrence relation: $L_N = L_x + \frac{1}{\frac{1}{L_r} + \frac{1}{L_{N-1}} }$ $L_0 = L_x +L_r$ If $L_x$ and $L_r$ are kept constant, I have no problem to find a solution to ...
0
votes
1answer
34 views

Construct a converging series that is substantially larger than a given converging series

This is more of a request for advice than a request for solution. Last night we were given the following and nobody figured it out in the time given (about 5 minutes). I think this is a problem many ...
3
votes
1answer
77 views

How prove $T_{n}\neq 0$,if $T_{n+2}=(1-2c)T_{n+1}+(2c+a-c^2)T_{n}-(a-c^2)T_{n-1}$

Question: Assmue that the postive integer $a,c$ ,such $\lfloor \sqrt{a} \rfloor=c$ ,Now let sequence $$y_{1}=1,y_{2}=-2c, y_{n+2}=-2c\cdot y_{n+1}+(a-c^2)y_{n},n\ge 1$$ show that ...
0
votes
1answer
9 views

Approximate average customer lifetime using churn

Average customer lifetime can be approximated using formula: $1/Churn$ (churn is a real number in interval $(0;1)$ The same can be achieved using more complex formula: $\sum_{n=0}^\infty ...
7
votes
1answer
304 views

Limit: How to Conclude

I have difficulty to conclude this limit ....; place of my attempts and results, can anyone help? tanks in advance $$\lim_{x\to +a}\, \left(1+6\left(\frac{\sin ...
0
votes
1answer
38 views

Sequence Covergent $\iff$ absolutely convergent

Is it correct that the sequence $a_n \rightarrow 0 $$\iff |a_n|\rightarrow 0$? I have it written down but I am unaware where I got it from. If so is it true for any limit?
3
votes
2answers
135 views

Closed form of a sum of binomial coefficients?

I have the following function: $T_n(d)=\sum\limits_{k=\frac{n-d}{2}}^{\lceil \frac{n}{2} \rceil}{k\choose \frac{n-d}{2}}$ ${n \choose 2k}$, where $n,d\in \mathbb{N}^0$, and $n,d$ have the same ...
3
votes
2answers
68 views

$\lim \sqrt[n]{a^n + b^n}$

I've seen some answers here for why this limit is the maximum between $a$ and $b$, but all of then included the hypotesis that $a$ and $b$ are both non negative. It was asked to show that this limix ...
1
vote
1answer
26 views

Build a sequence, $a_n$ with $PL=\{0,1,2\}$

I was asked to build a sequence which has exactly three partial limits: $\{0,1,2\}$. Also, for every $n\in\Bbb{N}: \left|a_{n+1} - a_n\right| < 1$ At first I thought about: $$a_n = \begin{cases} ...
2
votes
2answers
58 views

Convergence for all $\theta$ of a sum with periodic function

How can I show that: $$ \sum_{n \geq 1} \dfrac{\sin(n\theta)}{n} $$ converges for all $\theta \in \mathbb{R}$?
2
votes
1answer
56 views

Is the calculation of the series in this video correct?

I am watching this video (from MIT OCW) and Prof. Jerison is explaining about series. He is trying to calculate that if some blocks of equal length are kept on top of each other, will the last block ...
1
vote
2answers
39 views

Taylor series convergence for sin x

a. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}<\sin x<x-\frac{x^3}{3!}+\frac{x^5}{5!},$ b. $\forall x\in(0,\pi/2),\quad x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots-\frac{x^{4k-1}}{(4k-1)!}<\sin ...
1
vote
1answer
32 views

For what p is the series abolsutely convergent and conditionally convergent?

My lecturer has a passion for logs, and I'm reviewing some of her past papers and I found this question and I'm having a quite a difficult time dealing with, any help would be appreciated $$ ...
0
votes
2answers
52 views

convergence of sin functions absolutely for all x in $\mathbb R$ [closed]

How do I show that $\sum_{n = 1}^{\infty}2^n \sin(x/3^n)$ converges absolutely for all $x \in \mathbb{R}\;$?
9
votes
5answers
240 views

Prove that $\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx=\frac{16\sqrt{3}}{729}\pi^5+\frac{605}{54}\zeta(5)$

This integral comes from a well-known site (I am sorry, the site is classified due to regarding the OP.) $$\int_0^1\frac{1-x}{1-x^6}\ln^4x\,dx$$ I can calculate the integral using the help of ...
1
vote
1answer
85 views

Explanation of the formulas for sums $\sum nr^n$ and $\sum n^2 r^n$

So I am studying series for an exam right now and there is an example in the book I am studying (unfortunately the book is specific to my university so I cannot give any link) where certain series' ...
3
votes
2answers
65 views

Radius of convergence of power series which has factorial term

I am trying to find radius of convergence of the following power series: $\sum_{n\geq 1} n^n z^{n!}$ I tried ratio test but it became complicated, I have never seen such radius of convergence problem ...
1
vote
0answers
69 views

Why is sequence $(1+\frac{1}{n})^{n+1}$ descending? [duplicate]

I was studying the proof of $e$ number when I noticed something: Why is the sequence $(1+\frac{1}{n})^{n+1}$ descending? It starts ascending with grater n but in one moment it starts descending? Why ...
1
vote
1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
3
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0answers
49 views

Analytic solution for system of Digamma function equations

Peace be upon you, There is a while, I am seeking for the analytic solution of $\alpha$ and $\beta$ in this system \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ ...
0
votes
1answer
29 views

Writing dense sets in terms of set of integers

Can we write every dense set in $\mathbb R$ as {$x_n$}$\mathbb Z$ , where {$x_n$} is a real sequence with limit $0$ ?
2
votes
3answers
100 views

Show that $q^4+2pq^2 +p^2 = 2pq -(pq)^2 -1$ becomes $p^3+q^3+3pq-1=0$.

I know that these two are exactly the same equation but I can't seem to prove it. You are also given that $p+q=1$. This is a follow up from a similar question.
0
votes
1answer
58 views

Rate of convergence of an iterative root finding method similar to Newton-Raphson

We are defining an algorithm as follows: Let $f(x)$ be a function with a root in $[a,b]$. We define a series $\{x_k\}_{k=1}^{\infty}$ as follows: $x_{k+1}=x_k-f(x_k)\frac{b-a}{f(b)-f(a)}$. ...
2
votes
2answers
156 views

Prime Number Sum Sequence (Amateur)

SOLVED: This is false Beginning with 3, add the next consecutive prime (2) and then take that sum (5) and add that to next consecutive prime (3) to get (8), and so on... $$ 3 + 2 = 5 $$ $$ 5 + 3 = 8 ...
1
vote
0answers
62 views

sum of reciprocal power-1

I found this in my old notebook $$\sum_{n \text{ perfect power}} {\frac{1}{n-1}} = 1$$ and this was my "proof" $$ \begin{align} \frac{1}{1}+\frac{1}{2}+\cdots ...
1
vote
2answers
31 views

Divergence of a recursive sequence

If $(x_n)$ isthe sequence defined by $x_1=\frac{1}{2}$ and $x_{n+1}=\sqrt{x_n^2 +x_n +1}$, show that $\lim x_n = \infty$ Ive tried a couple of things but none of them helped. Ive tried to suppose, by ...
0
votes
1answer
25 views

Count of paths between N points

I am trying to arrive at a formula that will give me the number of distinct paths between a set of discrete points on a map. I have worked out that I can calculate it using a series of additions: ...
24
votes
1answer
534 views

Prove that $\displaystyle\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

Question. Let $f:[0,1]\to\mathbb R$ given by $$ f(x)=\left\{\,\,\, \begin{array}{ccc} \displaystyle{\left\lfloor{1\over x}\right\rfloor}^{-1}_{\hphantom{|_|}}&\text{if} & 0\lt x\le 1, \\ ...
0
votes
1answer
27 views

Sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$?

I've been trying to think of a simple example of a sequence of sets $f_n$ that converge almost everywhere to $f$ but not almost uniformly to $f$. Suppose $f$ is the zero function for the sake of ...
0
votes
2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
0
votes
5answers
57 views

Arithmetic Progression.

Q. The ratio between the sum of $n$ terms of two A.P's is $3n+8:7n+15$. Find the ratio between their $12$th term. My method: Given: $\frac{S_n}{s_n}=\frac{3n+8}{7n+15}$ ...