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

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2answers
20 views

A analytic representation of q- rational series

Using Mathematica, we can find $$\sum\limits_{n = 1}^\infty {\frac{{{{\left( {1 - q} \right)}^2}{q^n}}}{{\left( {1 - {q^n}} \right)\left( {1 - {q^{n + 1}}} \right)}}} = q,\;q \in \left( {0,1} ...
1
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0answers
32 views

Problem 14 from Baby Rudin chapter 3

Let $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1},\quad a_n=s_n-s_{n-1} \quad\text{for} \quad n\geqslant 1$$ Assume ...
10
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4answers
587 views

A way to calculate e?

Define three series: The first series is $$n^n: 1,\ 4,\ 27,\ 256,\ 3125,\ 46656, \ldots$$ The second series is that of the ratios between adjacent members of the first series, or ...
1
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1answer
126 views

Why is $1+2+3+\cdots = 0 $? [duplicate]

I had seen this result a while back in a Numberphile video: $1+2+3+\cdots = -\frac{1}{12}$ I was trying to prove the same result using a different method when I accidently proved that the sum was ...
2
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0answers
31 views

convergence of general harmonic series

My question is about determining the convergence of a general harmonic series using the integral test. According to the following resource: pg.32 , we can see that for a general harmonic series ...
2
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6answers
102 views

Find the limit of the sequence $(4^n)/((2n)!)$

How to find $$\displaystyle \lim_{n\to\infty} \frac{4^n}{(2n)!}$$ Can I use l'Hopital's rule?
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1answer
21 views

Dirichlet theorem

Can anyone give a simple number theory proof for the Dirichlet theorem? Statement of Dirichlet theorem:given any two numbers a and b whose g.c.d is 1,Prove that infinitely many primes exist in the ...
2
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2answers
49 views

How can I solve $\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$ step by step? [on hold]

$$\sum_{n=1}^{\infty} \frac{8}{(4n-1)(4n-3)}$$ First off I know the answer to this question, but I do not know how to actually get the answer I do know some basic sums like that one. Can someone ...
0
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1answer
65 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 ...
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1answer
47 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
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2answers
45 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
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2answers
41 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 Principal balance of a debt and constants $a$ and $b$ are constants or ...
0
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0answers
31 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 ...
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3answers
55 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
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1answer
59 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
52 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
17 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
35 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
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2answers
33 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
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3answers
58 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 + ...
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2answers
41 views

Prove that if $\lim_{n\to \infty} a_n=l$ then $\lim \sup a_n=\lim \inf a_n=l$

Let $(a_n)_{n\in\mathbb{N}}\subset\mathbb{R}$, a bounded sequence. For each $n\in\mathbb{N}$, we have $A_n=\{a_k:k\ge n\}$. Let $\lambda_n=\sup A_n$ and $\beta_n=\inf A_n$.So we have $(\lambda_n)$ and ...
4
votes
2answers
214 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)?
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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
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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
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1answer
24 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
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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 ...
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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 ...
12
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4answers
214 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
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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
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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 ...
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1answer
61 views
+50

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 ...
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1answer
40 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
71 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
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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 ...
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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
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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
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1answer
100 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
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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
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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
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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 ...
2
votes
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
448 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).
0
votes
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?