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

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0
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1answer
58 views

Prove sequence ${x_{n + 1}} = \sin {x_n},{x_1} = 1 $ has a limit

Prove that the sequence defined by $${x_{n + 1}} = \sin {x_n},\ {x_1} = 1$$ has a limit. Ok, I want to prove by Weierstrass: This sequence is monotonically decreasing Sequence is bounded ...
0
votes
1answer
40 views

high order (infinite series)

This question, I have made but there was no answer, so I will try again. If we have the sums $f(n) = 1^{59} + 2^{59} + 3^{59} + \cdots + (10^n)^{59}$ and $g(n) = 1^{5} + 2^{5} + 3^{5} + \cdots + ...
3
votes
2answers
32 views

Calculating in closed form another digamma alternating series

Is there any clever way of finish it fastly? $$\sum _{n=1}^{\infty } (-1)^{n+1} \left(\psi ^{(0)}\left(\frac{5}{8}+\frac{3 n}{8}\right)-\psi ^{(0)}\left(\frac{1}{8}+\frac{3 n}{8}\right)\right)$$ ...
1
vote
2answers
23 views

Explicit formula for recursive sequence.

Consider the series defined by $$P_n=(P_{n-1}-a)b\ .$$ $$P_0=c$$ Basically the number sequence $P_n$ represents the current Principle balance of a debt and constants $a$ and $b$ are constants or ...
0
votes
0answers
29 views

Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ with given conditions

If for any series $b(n)=2b(n-1)$ when $b(n)$ is odd number and $b(n)=b(n-1)$ if $b(n)$ is even number. then Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ Our Approach: I could not ...
1
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3answers
54 views

Why does this sum converge $\sum\limits_{k=1}^\infty\left (\frac{k\sin k}{2k+1}\right)^k$

I don't understand why this sum converges. $$\sum\limits_{k=1}^\infty \left(\frac{k\sin k}{2k+1}\right)^k$$ $$\lim_{x\to\infty} \left(\frac{k\sin k}{2k+1}\right) = diverge$$ I don't find any other ...
1
vote
1answer
51 views

Closed form of this sum

$$\sum _{ s=1 }^{ \infty }{ \left( \frac { 1 }{ 4s-1 } \sum _{ n=0 }^{ \infty }{ \left( \frac { 1 }{ n+1 } \sum _{ k=0 }^{ n }{ \left( \left( \begin{matrix} n \\ k \end{matrix} \right) \frac { { ...
3
votes
2answers
49 views

Sum involving zeta functions

Find closed form of the following - $$ \displaystyle \sum_{n=2}^{\infty}{\left(\frac{(n-1)\zeta(n)}{4n-1}\right)} $$ I don't know how to approach to it - Using the integral definition? I cannot use ...
0
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1answer
12 views

Summatory problem | Ordinary least square estimator

How I can transform the first expression in the second? \begin{align} \hat{\beta}_{1} & =\frac{n\sum X_{i}Y_{i}-\sum X_{i}\sum Y_{i}}{n\sum X_{i}^{2}-\left(\sum X_{i}\right)^{2}} \\ & = ...
0
votes
1answer
16 views

Piecewise $C_1$ and piecewise continuous

I would appreciate if the following questions could be clarified with your help. If a function is piecewise $C_1$, does this imply that it's also piecewise continuous? If a function is piecewise ...
0
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1answer
16 views

$f_n(x) = \left\lfloor \frac{\sin(2\pi (x / n + 1/ 4) + 1 }{2}\right\rfloor$ and related

$f_n(x) = \left\lfloor \frac{ \sin(2\pi (\frac{x}{n} + \frac{1}{4})) + 1}{2}\right \rfloor = 1 \iff x = kn$ and $ f_n(x) = 0 \iff x \neq kn$. Let $g_n(x)$ be what's within the floor brackets. Then ...
0
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0answers
34 views

sequence and their arithmetic means

Can it happen that $s_n>0$ and that $\limsup s_n=\infty$, although $\lim \sigma_n=0$ where $\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1}$. I want to take such sequence: $s_n$ is $\frac{1}{n}$ if $n$ is ...
1
vote
2answers
32 views

Cauchy's root test for series divergence

Just a question regarding determining the divergent in this example :$$\sum{ 1 \over \sqrt {n(n+1)}} $$ is divergent. It explains the reason by saying that $a_n$ > $1 \over n+1$. If I am not wrong it ...
3
votes
3answers
56 views

Calculate $\lim (\frac{1}{{1\cdot2}} + \frac{1}{{2\cdot3}} + \frac{1}{{3\cdot4}} + \cdots + \frac{1}{{n(n + 1)}})$

Calculate $$\lim \left(\frac{1}{1\cdot2} + \frac{1}{2\cdot3} + \frac{1}{3\cdot4} + \cdots + \frac{1}{n(n + 1)}\right) $$ If reduce to a common denominator we get: $$\lim \left(\frac{X}{{n!(n + ...
4
votes
2answers
212 views

How to compute fraction sums?

For example, $$\sum\limits_{k=1}^{n}\frac{1}{(2k-1)(2k+1)}=\frac{n}{2n+1}$$ Is there an easier way to evaluate fraction sums (without using partial sums)?
0
votes
1answer
30 views

What rotation rules can be applied to stacked cubes to make a 3D spirograph?

If you could arrange building blocks for example toy cubes so that every next cube was tilted over its base by 20 degrees and rotated to it's right by 15 degrees, it would form a helical structure. ...
0
votes
1answer
35 views

Division Alternating Series

In order to solve a Physics question, I was able to get to the point where I figured out that the answer was the sum of the following series: $$x = 42 + 14 + 2.8 + (14/15) + (14/75) + ...$$ As you ...
2
votes
1answer
22 views

Example of a pointwise convergent functional sequence that is not compactly convergent.

I'm looking for an example of a pointwisely convergent functional sequence $\{f_n\}_{n \geq 0}$ (where $f_n:\mathbb{R}\to\mathbb{R}$) that is not compactly convergent. I'm not sure if it is even ...
2
votes
2answers
54 views

How to determine a kind of distance between two permutations?

Let's define a distance between two permutation of length $N$: it is the minimum steps to change one to be another. "A step of change" means that exchanging any two elements' location. For example, ...
0
votes
1answer
22 views

Upper bound of the function

here you can read my first question on this topic, namely: $$\text{if } f\left(\frac{x}{3}\right)-f\left(\frac{x}{4}\right)\le Ax+B\ln x-C, $$ where $f(x)$ is my function and $A$,$B$,$C$ are ...
1
vote
3answers
25 views

Is this a counter example for a comparison test for sequences?

I’ve recently started learning about sequences and convergence and divergence, and I came across the comparison test for sequences. What I have is that: What if $a_n$ is defined as a periodic ...
10
votes
3answers
165 views

A conjectured result for $\sum_{n=1}^\infty\frac{(-1)^n\,H_{n/5}}n$

Let $H_q$ denote harmonic numbers (generalized to a non-integer index $q$): $$H_q=\sum_{k=1}^\infty\left(\frac1k-\frac1{k+q}\right)=\int_0^1\frac{1-x^q}{1-x}dx=\gamma+\psi(q+1),\tag1$$ where ...
0
votes
0answers
19 views

Deducing absolute convergence in particular cases from invariance under rearrangements

At least since the 19th century, it has been known that All sequences $\{a_n\}_{n=1}^\infty$ for which $\lim\limits_{n\to\infty} \sum\limits_{k=1}^n |a_k|<\infty$ are sequences ...
2
votes
1answer
44 views

How to prove that this series converges uniformly?

I have a series $$ -\frac{\pi}{12} + \sum_{k=1}^\infty \frac{\left(3k\pi^2-16\right)\sin{\frac{k\pi}{2}} + 8\pi\cos{\frac{k\pi}{2}}}{\pi^2k^3}\cos{kt} $$ And I have to use Weierstass test to prove ...
1
vote
1answer
41 views

A question about an infinite sequence of elementary row operations

Do there exist matrices $A$ and $B$ such that $B$ can be transformed into $A$ only if an infinite number of elementary row operations are performed on $B$? "What can we multiply the top equation by ...
0
votes
1answer
33 views

Graphic proof of an inequality between sequence ratios

I would like to verify my proof for the following claim. Let $b_i$ be a positive decreasing sequence, $j<k$ two integers and $d$ a positive number. Prove that: $$ ...
4
votes
3answers
64 views

Sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$

I've been working with the series: $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$$ From the ratio test it is clear that the series converges for $|x| < 1$, but I'm unable to obtain the sum ...
4
votes
5answers
76 views

Calculate $\lim_{n\to\infty} (n - \sqrt {{n^2} - n} )$

Calculate limit: $$\lim_{n\to\infty} (n - \sqrt {{n^2} - n})$$ My try: $$\lim_{n\to\infty} (n - \sqrt {{n^2} - n} ) = \lim_{n\to\infty} \left(n - \sqrt {{n^2}(1 - \frac{1}{n}} )\right) = ...
1
vote
1answer
47 views

Convergence of $\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$

I have trouble with the following sum: $$S=\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$$ Since $a_n=\frac{2}{n}$ decreases monotonically and tends to $0$, it converges by Liebniz criterion. Then, by ...
1
vote
4answers
66 views

How to prove that the following sequence will never contains number greater than 3

You may now the following sequence: 1 11 21 1211 111221 312211 13112221 Explanation of the sequence (I've put an hint just for the ones who want to search a bit ...
0
votes
0answers
20 views

Convergence of a sequence using Banach contraction principle: $x_{n+1}=\frac{1}{x_n+a},x_1>0,a \in \mathbb{R},n=1,2,…$

$$f(x)=\frac{1}{x+a}$$ $$f:\mathbb{R}_{\le0}\rightarrow \mathbb{R}_{\le0},a<0$$ $$f:\mathbb{R}_{=0}\rightarrow \mathbb{R}_{=0},a=0$$ $$f:\mathbb{R}_{\ge0}\rightarrow \mathbb{R}_{\ge0},a>0$$ ...
1
vote
1answer
99 views

Strange result about the log sum

I am working in a infinity sum and I get the strange result $$\sum _{n=1}^{\infty } \frac{1}{2} \log \left(\frac{1}{n^2}\right)=\log (2 \pi )$$ it seem as $$-2 \zeta '(0)$$ but i do not justify? it ...
3
votes
2answers
62 views

How is the Radius of Convergence of a Series determined?

Consider $$\sum_{n=0}^{\infty}\frac{(-1)^nx^n}{(n+1)^2}$$ which by the ratio test the ratio of two consecutive terms converges to $|x|$ as $n\rightarrow \infty$ and has a radius of convergence equal ...
1
vote
3answers
42 views

Evaluating the ratio $ {{a_{n+1}}\over{a_n}}$ in calculating the radius of convergence for a power series

In calculating the radius of convergence for the power series $$ \sum_{n=1}^\infty {{(2n)!}\over(n!)^2}\ x^n $$ By the ratio test, we let $$ a_n = \lvert {{(2n)!}\over(n!)^2}\ x^n \rvert \quad\quad ...
1
vote
0answers
37 views

Does the “alternating” harmonic series where only prime terms are negative converge?

We know that the harmonic series $\sum \frac{1}{n}$ diverges, yet the alternating harmonic series $\sum \frac{(-1)^n}{n}$ converges. Euler famously gave a proof of the infinitude (and of the ...
1
vote
3answers
63 views

Sum of $ \ \frac{1}{(\ln k)^{\ln k}} \ $

How do I find out if the infinite sum of $ \ \frac{1}{(\ln k)^{\ln k}} \ $ is convergent or divergent? I'm given a hint: $ \ \ln k \ = \ e^{\ln(\ln k)}$ but I can't figure out how to apply that.
0
votes
1answer
8 views

Equidistribution and Smaller Sets

I have a question about equidistribution. In Wikipedia, Equidistributed sequence shows in the equation in the definition that $n$ reaches almost infinity. I was just wanting to make sure if ...
2
votes
1answer
26 views

How many unique numbers can be obtained by adding two numbers from two different sequences?

Let the two integer sequences $\{a_m\}$ and $\{b_m\}$, be defined as: $a_n+D_n=a_{n+1}$ and $b_n=a_n-k$, where $D_n$ may be any natural number (and $D_i$ may or may not be equal to $D_j$), $k$ is an ...
5
votes
3answers
443 views

What is the pre-requisite knowledge for generating my own integer sequence?

I've recently come across the On-Line Encyclopedia of Integer Sequences and I'm completely fascinated by it; something about how easy integers are to grasp and yet how complex the sequences are. I ...
1
vote
3answers
53 views

Geometric sequence problem

Determine the value(s) of k, so that the positive numbers $\log_8(k-1)$, $3\log_8(k-1)$ and $6$ form a geometric sequence (in order given above).
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3answers
34 views

Arithmetic and geometric sequence

Which two numbers should be placed between -5 and 49 so that the first three numbers form an arithmetic sequence, whereas the last three numbers form a geometric sequence?
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votes
2answers
67 views

Is the Taylor series of the $f: \mathbb{R} \to \mathbb{R}$ evaluated at $0$ converges pointwise (in the whole $\mathbb{R}$)?

I would like to solve this problem: Consider the $f: \mathbb{R} \to \mathbb{R}$ function which is $n$-times differentiable for any $n \in \mathbb{N}$. Is it true that the Taylor series of ...
1
vote
2answers
77 views

Prove $\lim \frac{3^n + 2^n}{5\cdot3^n + 7\cdot2^n} = \frac{1}{5}$

Prove using limit definition. $$\lim \frac{3^n + 2^n}{5\cdot 3^n + 7\cdot 2^n} = \frac{1}{5} $$ My try: $$\left| {\frac{3^n + 2^n}{5\cdot 3^n + 7\cdot 2^n} - \frac{1}{5}} \right| < \varepsilon ...
1
vote
1answer
19 views

Meaning Behind Mapping from a Compact Subset to Another Set

Suppose I tell you that a set, $A$, is compact and a subset of a metric space. This means that it is closed and bounded and that every sequence in set $A$ has a converging sub-sequence. Then I tell ...
0
votes
2answers
28 views

finding for which $c\in \mathbb{R}$ sequence converges

so i am trying to find for which $c\in\mathbb{R}$ this sequence converges: $a_{1}=c$ and $a_{n+1}=1+\frac{a_{n}^2}{4}$ So i got the basic idea how to do this. First i found the candidate for limit: ...
1
vote
3answers
37 views

Product of first $n$-th prime power integers $+ 1$

I was just playing with prime numbers and then I accidentally found this pattern. Let $p_1\cdot p_2\cdot p_3\cdots p_n$ is the product of first $n$-th prime power integers. Prove that: $p_1\cdot ...
0
votes
4answers
87 views

About the limit $\lim_{n\to +\infty}\frac{n^k}{n!}$ for a fixed $k\in\mathbb{N}$.

Given a natural number $k$ and some real number $\epsilon>0$, I have to prove that there exists a natural number $n$ such that $\frac{n^k}{n!}<\varepsilon$. I tried to develop for ...
2
votes
3answers
315 views

Which is the limit of the sequence $\sum_{i=1}^n \frac{\cos(i^2)}{n^2+i^2}$

I can't find the limit as n $\to$ infinity of the sequence: $$\frac{\cos(1)}{n^2 + 1} + \frac{\cos(4)}{n^2 + 4} + \dots + \frac{\cos(n^2)}{n^2 + n^2}$$ I tried to use the inequality $\cos(n^2) < ...
4
votes
0answers
74 views

Limit of $\frac{1-2+\cdots+(2n-1)-2n}{\sqrt{ (n^2+1)}+ \sqrt{ (n^2-1)}}$

Here is the limit to be calculated : $$\lim_{n\to\infty}\frac{1-2+\cdots+(2n-1)-2n}{\sqrt{ (n^2+1)}+ \sqrt{ (n^2-1)}}$$ The question provides no other information. Doubts What is the domain of ...
1
vote
1answer
33 views

Inverse sum representation of sine

The other day I was playing with functions of the form $$ f(x) = \frac{1}{\frac{1}{a_0(x-b_0)} + \frac{1}{a_1(x-b_1)} + \cdots + \frac{1}{a_n(x-b_n)}} $$ and I found particularly that $$ ...