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

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11
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5answers
151 views

Evaluate $\frac{ 1 }{ 1010 \times 2016} + \frac{ 1 }{ 1012 \times 2014} + \frac{ 1 }{ 1014 \times 2012} + \cdots + \frac{ 1 }{ 2016 \times 1010} = ?$

$$\dfrac{ 1 }{ 1010 \times 2016} + \dfrac{ 1 }{ 1012 \times 2014} + \dfrac{ 1 }{ 1014 \times 2012} + \cdots + \dfrac{ 1 }{ 2016 \times 1010} = ? $$ My attempt so far : ...
3
votes
1answer
179 views

Non- linear recurrent relation (exponential term)

Is there any solution to this recurrent relation: $X_n=\alpha-e^{-\beta X_{n-1}}$, $X_0=0$, $\alpha>1$ and $\beta>0$
1
vote
0answers
93 views

Has anyone seen this form of the Collatz Conjecture?

This question asks if this form of the Collatz Conjecture has been reported or is all ready known. The goal of this question is to determine if I should write a paper on it's discovery or not and ...
3
votes
3answers
63 views

Convergence of a series with alternating denominator - Real Analysis

Decide if the series converges absolutely, conditionally, or not at all. \begin{equation} \sum_{n=1}^{\infty}\frac{(-1)^n}{(2+(-1)^n)n} \end{equation} I'm having a lot of trouble with this one. I ...
2
votes
1answer
120 views

A closed form for $\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $

This is another 'arctanlog' series: $$ S=\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $$ Maybe differentiating with respect to some parameter could be of interest. What ...
2
votes
2answers
51 views

Sequence vs Series

What is the difference between a sequence and a series and how should they be used i.e. give examples of the usage of these terminologies in separate senarios.
0
votes
1answer
30 views

Convergence uniformly implies in integral

Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ such that ...
2
votes
1answer
69 views

Prove that $\frac {1}{a_1a_2} + \frac {1}{a_2a_3} + \frac {1}{a_3a_4} + … + \frac {1}{a_{n-1}a_n} = \frac {n-1}{a_1a_n}$

There is this question in one of my math textbooks which I can't seem to figure out how to solve, it'd be awesome if you could help me : If $a_1,a_2,a_3,...,a_n$ IS an arithmetic progression and ...
0
votes
1answer
28 views

Correlation between convergence radius of complex series

What do we know about the convergence between complex power series that look almost the same? For instance, if we have series $\sum_{n=1}^{\infty}a_n z^n,$ $\sum_{n=1}^{\infty}a_n z^{n+1},$ and ...
0
votes
0answers
24 views

Resource for learning sequences and series.

By reading various questions and answers on this site, I realized I'm particularly weak at sequences and series, for example various tests for convergence tests, Cauchy sequences and so on. It's ...
0
votes
1answer
44 views

Show that a subspace of a normed vector space is closed

Let $X$ be a normed vector space over $\mathbb K, \mathbb K = \mathbb R$ or $\mathbb K=\mathbb C.$ Let $Y$ be a closed linear subspace of $X$ and $x\in X\backslash Y.$ Set $Z=\{y+\alpha x;\;y\in ...
1
vote
0answers
52 views

Can one prove the divergence of $\sum \frac{1}{p}$ by the absolute convergence criterion of infinite products?

Euler proved this celebrated theorem that $\sum \frac{1}{p}=\infty$ by using the product formula that $\displaystyle \zeta (s)=\prod \left( \frac{1}{1-p^{-s}}\right)$. Now I thought of another ...
0
votes
3answers
59 views

Assume $a_1 > 1$. Find the limit of $a_{n+1} = 2 - ( \frac{1}{a_n} + a_n )$.

Attempt: $$L = 2 - \frac{1}{L} $$ By solving this, I can get that $L=1$. However, in trying to prove the limit existence, I want to do this by proving the recursive sequence is bounded and ...
0
votes
1answer
32 views

Stuck trying to find unbounded $s_n$ with $\frac{1}{n}\sum_{k=1}^n s_k\rightarrow L$

I proved that if a sequence $(s_n)$ converges to a limit $s$ then so does its "average sequence," $(\sigma_n)$ with $\sigma_n=\frac{1}{n}\sum_{k=1}^n s_k$. I found a counterexample for the converse, ...
3
votes
2answers
37 views

Constructing a sequence with subsequence

Construct a sequence $(x_n)_{n=1}^\infty $ such that for every real number L, there is a subsequence $(x_{n_k})_{k=1}^\infty $ with $\lim_{n\rightarrow \infty} x_{n_k}=L$ I don't really know how to ...
2
votes
3answers
72 views

Show that limit of sequence does not exist

Problem: Let $x_n=\begin{cases}\frac 1n &\text{if $n$ is odd} \\ 1 & \text{if $n$ is even} \end{cases}$ and show that the limit $\lim_{n \to \infty} x_n$ does not exist. I am trying to ...
1
vote
1answer
52 views

Proving a Special Case of a Limit Theorem

I'm having trouble proving a special case of the limit theorem below. I attempted a proof by contradiction that appears to me to make sense in the first direction but I'm not able to come up with ...
2
votes
2answers
117 views

Sum the series (real analysis)

$$\sum_{n=1}^\infty {1 \over n(n+1)(n+2)(n+3)(n+4)}$$ I tried to sum the above term as they way I can solve the term $\sum_{n=1}^\infty {1 \over (n+3)}$ by transforming into ${3\over n(n+3)} ={1\over ...
1
vote
3answers
51 views

Convergent and divergent series with root test

Find a convergent series $\sum_{n=1}^\infty a_n$ with positive entries, such that $\lim_{x\to \infty}\sqrt[n]{a_n}=1$ Find a divergent series with similar properties. Any hints on that? Thanks in ...
0
votes
0answers
40 views

Show that $x(\pi - x)= \frac{\pi^2}{6}-\sum_{k=1}^{\infty} \frac{\cos(2kx)}{k^2}$

Show that $$x(\pi - x)= \frac{\pi^2}{6}-\sum_{k=1}^{\infty} \frac{\cos(2kx)}{k^2}$$ for $ 0<x<\pi$ My idea: I've defined the periodic function $$f(x) = 0 \text{ if } x \in [- \pi, 0) \text{ ...
2
votes
2answers
85 views

Infinite Sum of the Reciprocal of the Inverse Harmonic Number Function

It's a very strange question I know, but I am trying to figure out the infinite series $\sum_{k=1}^\infty \frac {1}{H^{(-1)}_k}$ where $H^{(-1)}_k$ is the inverse of the harmonic number function ...
1
vote
2answers
34 views

Summing series with factorials and fractions

I know that $f(r) = r!$ and $f(r+1) - f(r) = r \cdot r!$ But I have no idea how to apply that to fractions. I have a question, show that $\dfrac{r-1}{r!} = \dfrac{1}{(r-1)!} - \dfrac{1}{r!}$ and I ...
1
vote
2answers
44 views

Properties of increasing sequences

Suppose that a sequence {${x_k}$} is increasing. Show that there is some $N$ such that either $x_k \geq 0$ for all $k \geq N$ or $x_k \lt 0$ for all $k \geq N$. That is, eventually its terms are all ...
0
votes
0answers
19 views

Determining convergence of series of form $\sum a_n z^{n^k}$

So, title says it all. Say I have a (complex) series of the form $$\sum_{n=0}^\infty a_n z^{n^k},$$ for some $k\in \mathbf{N}$. I'm a little at loss what to do with it to determine its radius of ...
2
votes
3answers
71 views

Why do we set $u_n=r^n$ to solve recurrence relations?

This is something I have never found a convincing answer to; maybe I haven't looked in the right places. When solving a linear difference equation we usually put $u_n=r^n$, solve the resulting ...
0
votes
3answers
45 views

Arithmetic and geometric progression question (too long for title)

Good evening, I've been struggling with this one for a while now. The sum of an increasing geometrical progression is equal to 65. If we substract 1 from the smallest number there, and 19 from the ...
1
vote
3answers
48 views

What sequences where the difference between their consecutive terms is always a fibonacci numbers?

What sequence where the difference between its consecutive terms is always a fibonacci numbers ? I am trying to figure out a pattern in this sequence : 1,2,4,7,12,20,33,54,88
2
votes
1answer
39 views

Confusion about a certain series expansion

While reading some old notes on contour integration, I noticed the author uses series expansion: $$\frac{\sinh sx}{\sinh ...
0
votes
1answer
28 views

Is there a general formula for the following expression?

I am working on a proof, and came to the point where I need a general expression for the formula, taking real numbers $x_i$ and an integer $k$, of : $(x_1+x_2+...+x_n)^k$ I now I could apply the ...
0
votes
1answer
58 views

Generating function for a binary sequence

I don't know this subject so my question may not be expressed in the accurate form. Is there a function, or a structure, that generates any desired sequence of 0 and 1 of length n? Assume we can pad ...
0
votes
1answer
63 views

Calculating fourier series

I've a fourier series with a period = $2\pi$ that is even. f(t) = \begin{cases} 0 \text{, when: } 0<t<\pi-2 \\ \pi \text{, when: } \pi-2<t<\pi \end{cases} The functions trigonometric ...
2
votes
2answers
33 views

Cauchy property of a series

Are these two definitions equivalent, even though the first one has an extra term: If we consider the series $\sum_{n=1}^{\infty}x_{n}$ and the formal definition of a Cauchy property defined in terms ...
1
vote
1answer
57 views

Make a one sequence

A sequence of integers is a one- sequence if the difference between any two consecutive numbers in this sequence is -1 or 1 and its first element is 0. More precisely: a1, a2, ..., an is a ...
7
votes
3answers
73 views

Bound for $\sum_{k=1}^\infty\left(\frac{1}{2^k+k^2}\right)$

I found for the series: $$S=\sum_{k=1}^\infty\left(\dfrac{1}{2^k+k^2}\right)$$ a bound: $$S\le\dfrac{\pi^2}{6+\pi^2}$$ which is in good agreement with the approximate value of $S$ calculated with ...
0
votes
1answer
41 views

Proportional progression with limit.

In fact my problem more about finding the name of the Math topic that I have to study to solve my real problem. I'm going to explain it so maybe you can help me. I have an area of 100cm. Inside this ...
2
votes
4answers
100 views

Can I claim that $\sum _n ^{\infty }a _n=\sum _n ^{\infty }a _n^++\sum _n ^{\infty }a _n^- $

How can I motivate the following. I'm writing a proof for that if a seris $\sum a _n $ converges conditionally then the series consisting of its negative terms and the series consisting of its ...
2
votes
1answer
57 views

Is this series absolutely convergent?

Does the series $\displaystyle\sum_{n=1}^\infty(-1)^{n+1}\frac{1}{n(\ln n)^2}$ absolutely converge? The index n=1 kept me hanging too. This is an item I saw in one of my books. Isn't the series wrong ...
1
vote
2answers
40 views

$\lim_{n \to\infty} p_n = p$ implies $\lim_{n \to \infty}p_n^3 = p^3$

In an example from my lecture notes, I have that if $( p_n )_n$ is a sequence and $\displaystyle\lim_{n \to \infty} p_n = p$, then $\displaystyle\lim_{n \to \infty} p_n^3 = p^3$. I don't understand ...
3
votes
2answers
103 views

Series question,related to telescopic series, 1/2*4+ 1*3/2*4*6+ 1*3*5/2*4*6*8 …infinity [duplicate]

The series is $$\frac{1}{2*4}+ \frac{1*3}{2*4*6}+ \frac{1*3*5}{2*4*6*8}+....$$ It continues to infinity.I tried multiplying with $2$ and dividing each term by$(3-1)$,$(5-3)$ etc,starting from the ...
3
votes
1answer
108 views

Recursive sequence $a_{n+1} = \sqrt{a_1 + … + a_n}$

I need a hint to solve this: Let $a_1 = 1$ and define a sequence recursively by $$a_{n+1} = \sqrt{a_1 + a_2 + ... a_n}$$ Show that $$\lim_{n \to \infty} \dfrac{a_n}{n} = \dfrac{1}{2}$$ Any help? ...
1
vote
1answer
44 views

Train distances leaving at certain times

A train leaves Boston to Fort Lauderdale traveling at $125$ mph. An hour later, another train leaves Fort Lauderdale traveling to Boston at a rate of $140$ mph. When the two trains meet each other, ...
0
votes
1answer
50 views

$a_{2014}$ sought in a sequence

$a_n$ sequence is defined as follows: $a_1=0$ and $a_n$ is the smallest positive whole number, such that within $a_1,a_2,...,a_{n-1},a_n$ there is no arithmetic progression of 3 terms. How do I ...
0
votes
1answer
92 views

What comes next? For 8 year olds

This question is from the homework of my niece. She is 8 years old. And I could not help her with this question. There are 5 x 3 cells. And there is a number in each cell. Problem asks what should be ...
0
votes
2answers
145 views

Does a Sequence Converge to 0 if and only its Reciprocal Sequence Diverges to Infinity?

I was considering yesterday whether or not the question in the title is, in fact, true. I believe that the definition of a convergent sequence is $(\forall \epsilon>0)(\exists ...
1
vote
1answer
45 views

Can one express in a compact form a double, triple, etc., alternating series?

Trivially, a series like$\ a_0-a_1+a_2-a_3+...$ can be written as$$\ \sum_{i=0}^\infty (-1)^ia_i.$$ But what if I want to rewrite$\ a_0+a_1-a_2-a_3+a_4+a_5-...$ ,$\ a_0+a_1+a_2-a_3-a_4-a_5+...$ and so ...
0
votes
2answers
39 views

Rewrite $\sum_{n\ge0}{z^{kn}}$ to $\sum_{n\ge0}{f(n,k)z^n}$

In the simplest case: $\sum_{n\ge0}{z^{2n}}=\sum_{n\ge0}{\frac12((-1)^n+1)z^n}$. How to express $f(n,k)$ in closed form? If it's intractable, how to avoid piecewise expression?
3
votes
2answers
105 views

Closed-form for $\sum _{k=4}^{\infty }{\frac { \left( -\ln \left( 2 \right) \right) ^ {k}\zeta \left( 4-k \right) }{k!}}$

Can anybody find a closed form for this infinite sum? $$S = \sum _{k=4}^{\infty }{\frac { \left( -\ln \left( 2 \right) \right) ^ {k}\zeta \left( 4-k \right) }{k!}},$$ where $\zeta$ is the Riemann ...
1
vote
2answers
67 views

Partial sum formula of a polynomial series?

I am trying to find the partial sum formula of the following series: $$ \sum_{y=1}^{\infty} \frac{4y^2-12y+9}{(y+3)(y+2)(y+1)y} $$ I have tried using Faulhaber's formula without success. I have also ...
4
votes
0answers
91 views

Closed form of arctanlog series

What tools would you recommend me for $$\operatorname{arctan}\left( \frac{1}{1}\right)\log\left(1+\frac{1}{1}\right)+\operatorname{arctan}\left( ...
1
vote
2answers
62 views

Sequence of alternating $0$'s and $1$'s in terms of $i$?

How to redefine the function $f(n) = \begin{cases} 1, & \text{if $n$ is even} \\ 0, & \text{if $n$ is odd} \end{cases}$ in terms of arithmetic operations using ⅈ?