For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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2
votes
3answers
68 views

How do you evaluate this sum of multiplied binomial coefficients: $\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} $?

We have to find the value of x+y in: $$\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} = \binom{x}{y} $$ My approach: I figured that the required summation is nothing but the coefficient of $x^3$ is the ...
2
votes
2answers
35 views

Comparision test for this series?

How do I check divergence of this series? $$\sum_{n=0}^{\infty} \frac{6}{4n-1} - \frac{6}{4n+3}$$ Wolframalpha said it used the comparision test but I don't see what possible smaller sum to use? ...
2
votes
1answer
50 views

conditional convergence of $\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$

prove that the series $$\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$$ is conditionally convergent? I tried to prove that it is not absolutely convergent series by trying to prove that $\sum_{n=2}^{\infty} \...
3
votes
1answer
129 views

Series involving the Riemann zeta function

Consider the series: $$\sum_{n=1}^{\infty}\frac{\zeta(2n+1)}{n(2n+1)}$$ We can easily prove that it's a convergent series. My question, is there a way to express this series in terms of zeta ...
5
votes
0answers
265 views

An interesting identity involving powers of $\pi$ and values of $\eta(s)$

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
0
votes
1answer
28 views

Asymptotically equivalent series for uniform convergence

I have to find sets of uniform convergence of $$\sum_{n=1}^{\infty}n^2 \sin \frac{x}{n^4}$$ what if I study this series passing to the asymptotically equivalent $$\sum_{n=1}^{\infty}n^2 \frac{x}{n^4}$$...
1
vote
1answer
64 views

Let$\{a_n\}_{n=0}^{\infty}$ be bounded. Then $\sum_0^{\infty} z^na_n$ converges for $|z|<1$

How to prove: Let $\{a_n\}_{n=0}^{\infty}$ be bounded. Prove $\sum_0^{\infty} z^na_n$ converges for $|z|<1$ So far I prove that the partial sums form a Cauchy sequence, i.e. $\{S_k\}_{k=...
1
vote
0answers
33 views

Cesàro's Lemma - precise definition of limit, indices

Probability with Martingales: I'm a little confused about the precise langauge. I guess we have that $$\forall M > 0, \exists N_b > 0 \ \text{s.t.} \ b_n > M \ \text{whenever} \...
2
votes
1answer
34 views

Relationship between Cesàro's Lemma and Stolz–Cesàro theorem

Probability with Martingales: From Wiki: What is their relationship? Does any imply the other?
0
votes
0answers
22 views

Using ratio test for sequences?

Don't mark this as duplicate. The question is to verify whether my method is correct. Prove: $$\lim_{n\to\infty} \frac{x^n}{n!} = 0$$ Method: Let $\sum_{n=1}^{\infty} \frac{x^n}{n!}$. By ratio ...
4
votes
1answer
68 views

Calculate a limit $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \Big\{\frac{k}{\sqrt{3}}\Big\} $

The problem is to calculate a limit $$ \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \Big\{\frac{k}{\sqrt{3}}\Big\} $$ where {$\cdot$} is a fractional part. I believe that this limit is equal to $\...
1
vote
1answer
38 views

is the sevenacci sequence the only generalization of the fibonacci sequence that has no odd prime terms?

This question came up recently. Obviously the answer is no, but how can I know for sure without checking a whole bunch of generalizations? $s_1 = ... = s_6 = 0, s_7 = 1, s_n = s_{n-1} + ... + s_{n-...
4
votes
6answers
119 views

Prove that $ 1+2q+3q^2+…+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $

Prove: $$ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $$ Hypothesis: $$ F(x) = 1+2q+3q^2+...+xq^{x-1} = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} $$ Proof: $$ P1 | F(x) = \frac{1-(...
1
vote
1answer
22 views

Complete basis for space of summable monotonically decreasing sequences

I'm trying to extract a response function out of some input and output data. This response function is a sequence of values $r_n$ with $n\in\mathbb{N}$. There is a good reason why $r_n$ should be ...
19
votes
0answers
301 views

How would you evaluate $\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$

How would you evaluate the limit inferior of the sequence $n\,|\mathopen{}\sin n|$? That is, $$\liminf\limits_{n\to\infty} \ n \,|\mathopen{}\sin n|$$ Edit. Let $\mu$ be the irrationality measure ...
27
votes
2answers
459 views

Convergence of sequence: $ \sqrt{2} \sqrt{2 - \sqrt{2}} \sqrt{2 - \sqrt{2 - \sqrt{2}}} \sqrt{2 - \sqrt{2 - \sqrt{2-\sqrt{2}}}} \cdots $ =?

In other words, if we define a sequence $$ \displaystyle a_{n+1} = \sqrt{2-a_n}, \,\,\,a_0 = 0 .$$ Then, we need to find $$ \displaystyle \prod_{n=1}^{\infty}{a_n}. $$ Well, from here I don't seem ...
1
vote
0answers
22 views

Monotonic vector linear trasformed to another monotonic vector

Let $b = (b_1,\ldots,b_n)\in \mathbb{R}^n$ such that $\forall i \in \left\{ 1,\ldots, n \right\}$ we have $b_i > 0$ and $\forall i \in \left\{1,\ldots,n-1 \right\}$ we have $b_i > b_{i+1}$. Let ...
1
vote
0answers
25 views

If $s_n\leqslant t_n$. What we can say about their upper limits?

Suppose that $s_n\leqslant t_n$ for all $n\in \mathbb{N}$. We know that $\liminf \limits_{n\to \infty}s_n\leqslant \liminf \limits_{n\to \infty}t_n$. What we can say about their $\limsup$? Is it ...
0
votes
0answers
50 views

Seeking general formula for Euler Sum $\sum\limits _{ n=1 }^{ \infty }{ \frac {{\left({H}_{n}\right)}^{p}}{{n}^{q}}}$ with $q$ even and $p$ odd

I am wondering if there is a formula for $\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { { \left( { H }_{ n } \right) }^{ p } }{ { n }^{ q } } }$ with 'q' being an EVEN positive integer and 'p' ...
1
vote
1answer
39 views

Number of unique combinations

I have a collection of possibly repeated items that I need to map onto another set with possibly repeated items. I need to know an efficient algo to iterate through the unique mapping combinations. ...
0
votes
0answers
35 views

Check whether number belongs to recurrence relation

the blog article mentions a way to determine whether a given number $N$ is a Fibonacci number or not (Gessel criteria): $N$ is a Fibonacci number iff either $5N^2 + 4$ or $5N^2-4$ is a perfect square. ...
3
votes
3answers
87 views

How do I find $\displaystyle\lim_{n\to \infty}\left [\left (1+\frac{2}{n^a}\right )^{-n^b}n^c\right ]$ for real $a,b,c$ and $n\geq 1$?

I am not sure how to do this but the hint given is to consider various cases of $a>b$, $a<b$, $a=b$ and $c>0$, $c<0$.
1
vote
1answer
37 views

Proof of convergence of an infinite sequence

A question that I tried to prove is as follows. "Consider the following sequence defined recursively by $a_1=\sqrt{a}$ and $a_{n+1}=\sqrt{a+a_n}$, where $a>2$. (The first few terms are: $\sqrt{a}, ...
3
votes
2answers
46 views

Does a sequence that is eventually constant have less terms than one that is not?

Does a sequence that is eventually constant contain less terms than one that is not? I don't know how to properly think about this, one could either argue that a sequence : 1 2 3 1 1 1 1 1 1 1 1 ...,...
1
vote
1answer
55 views

How to compute equation with exponents?

I want to find $a$, where that term satisfies this equation: $$a + a(1-a) + a(1-a)^2 + \cdots + a(1-a)^{15} = 0.5$$ I could write that as a sum from 0 to 15, but still it is unclear to me how should ...
0
votes
2answers
36 views

The sum of the perimeter of regular polygons inscribed inside of regular polygons

This is a question combining number theory and geometry. I am asking it purely from curiosity, but I think it might be a useful and interesting question. Start with an equilateral triangle of ...
2
votes
2answers
72 views

Prove $\sum \frac{\cos nz}{n!}$ converges on compact sets. [closed]

Prove that $$\sum \displaystyle\frac{\cos nz}{n!}$$ converges on compact subsets of complex plane.
7
votes
4answers
103 views

Proof of Leibniz $\pi$ formula

I found the following proof online for Leibniz's formula for $\pi$: $$\frac{1}{1-y}=1+y+y^2+y^3+\ldots$$ Substitute $y=-x^2$: $$\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots$$ Integrate both sides: $$\...
1
vote
1answer
60 views

A sequence $a_{n+1}=\frac{a_n}{a_n^2+1}$

A sequence $\left(a_n\right)$ of real numbers is defined recursively: $$a_1>0, a_{n+1}=\frac{a_n}{a_n^2+1}$$ Prove that there exists an positive integer $n$ for which $$a_n>\frac{7}{10\...
8
votes
2answers
243 views

Limit of the sequence $a_{n+1}=\frac{1}{2} (a_n+\sqrt{\frac{a_n^2+b_n^2}{2}})$ - can't recognize the pattern

Consider the sequence: $$a_0=x,~~~b_0=y$$ $$a_{n+1}=\frac{1}{2} \left(a_n+\sqrt{\frac{a_n^2+b_n^2}{2}} \right),~b_{n+1}=\frac{1}{2} \left(b_n+\sqrt{\frac{a_n^2+b_n^2}{2}}\right)$$ $$\lim_{n \to \...
0
votes
1answer
19 views

Random subseries of harmonic series expected to converge, but how often does it?

Inspired by a previous question which I can't seem to find, what if we have $$X = \sum_{k=1}^{\infty}\frac{1}{k}\cdot P\left(U(0,1)<\frac{1}{k}\right)$$ That is, each term of the series will be $\...
0
votes
0answers
19 views

Summary of results concerning interchange of limits in series

The document http://www2.iugaza.edu.ps/ar/periodical/articles/volume%2014-%20Issue%201%20-studies%20-16.pdf constructs the theory of double sequences and double series very nicely, supplying the ...
0
votes
1answer
27 views

Is it true that $(\Phi_{-t})_*w\circ \Phi_t=\sum_{n=0}\frac{t^n}{n!}\mathcal L^n_vw$?

Let $v,w$ be vector fields and let $\Phi_t$ be the flow generated by $v$, that is $\frac{d}{dt}\big|_0\Phi_t(x)=v(x)$. The Lie derivative of $w$ in direction of $v$ is usually defined as $\mathcal ...
13
votes
5answers
2k views

Are we guaranteed that the harmonic series minus infinite random terms always converge?

Consider the known harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ and modify it as follows $$\sum_{n=1}^\infty a_n\frac{1}{n}$$ where $$a_n \sim \operatorname{Bern} \left({\frac{1}{2}}\right)$$ i.e. ...
0
votes
1answer
27 views

Nested Hypergeometric series

Is it possible to express the following series as a hypergeometric function: $$\sum_{n=0}^\infty (a)_n \sum_{j_1+j_2+\cdots+j_k=n} \frac{1}{(b)_{j_1} (b)_{j_2}\cdots (b)_{j_k}} z^n $$ where $(a)_n, (...
0
votes
2answers
33 views

Two sequences, one of them bounds difference of the other and converges to $0$. Show that the other sequence converges.

That is, let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be sequences such that $b_n \overset{n \to \infty}{\to} 0$ and for all $k \in \mathbb{N}$ and $l \geq k$, $$|a_l - a_k| < b_k\text{.}$...
0
votes
1answer
32 views

Infinite sum of partial sums

If, for an infinite sequence $a_1, a_2, a_3...$ we say that the partial sum is: $$\sum_{i}^n a_i$$ could the sum of the partial sums converge (retaining a constant first term, or $a_1$? That is, is ...
1
vote
0answers
51 views

Finding hte value of $1/(1^2)+1/(2^2)+1/(3^2)+\dots +1/(n^2)$ [duplicate]

Sorry, my English isn't quite well. Here is the problem. When $n$ moves toward infinity $$ 1/(1^2)+1/(2^2)+1/(3^2)+\dots +1/(n^2)=? $$ Any ideas?
0
votes
2answers
91 views

Find $\lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}$

I have problem with finding the following limit. I suspect that it should be easy, but really I don't have a clue. $$ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2} $$
0
votes
0answers
65 views

Formula for $\pi$ using primes

In one of his videos (https://www.youtube.com/watch?v=HrRMnzANHHs), Matt Parker introduces the following formula for $\pi$ using primes: $$\left(1-\frac{1}{3}\right)\cdot\left(1+\frac{1}{5}\right)\...
0
votes
1answer
16 views

Cauchy with terms of differing signs after some term $\implies$ convergence

Let $(x_n)_{n=1}^{\infty}$ be Cauchy. For each natural number $N$ (natural numbers starting at $1$), suppose there is an $n_1 \geq N$ and $n_2 \geq N$ such that $x_{n_1} < 0$ and $x_{n_2} > 0$. ...
2
votes
1answer
61 views

Series expansion of $\int x^xdx$

The indefinite integral: $$J=\int x^xdx$$ has no known closed form solution. Expanding in series the function $f=x^x$ we get: $$f\simeq\sum_{k=0}^N \dfrac{x^k\ln(x)^k}{k!}$$ So we can write: $$J\...
3
votes
1answer
86 views

Limit and rate of convergence of the sequence $a_{n+1}=\frac{a_n^2+b_n^2}{a_n+b_n},~~b_{n+1}=\frac{a_n+b_n}{2}$

Define the sequence the following way for some $x,y \geq 0$: $$a_0=x,~~~~~~~b_0=y$$ $$a_{n+1}=\frac{a_n^2+b_n^2}{a_n+b_n},~~~~~~b_{n+1}=\frac{a_n+b_n}{2}$$ Obviously: $$a_n \geq b_n,~~~~n \geq 1$$ ...
2
votes
1answer
62 views

Prove an algorithm for logarithmic mean $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{a_0-b_0}{\ln a_0-\ln b_0}$

Take: $$a_0=x,~~~~b_0=y$$ $$a_{n+1}=\frac{a_n+\sqrt{a_nb_n}}{2},~~~~b_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2}$$ Then we obtain as a limit the logarithmic mean of $x,y$: $$\lim_{n \to \infty} a_n=\lim_{n ...
0
votes
0answers
10 views

Composition of limits of functions | Switching limits of function

I have a question which I am having some trouble with. I have a double indexed sequence of stochastic processes (martingales in fact), denoted $X_{m,n}(t)$. Now I can prove that $\underset{m \...