Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: recurrence relations, convergence tests, identifying sequences, identifying terms. For questions on finite sums, use the (summation) tag instead.

0
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0answers
29 views

An equality about infinite sums involving squared harmonic numbers

I found the following equality. But I forgot the source as well as the solution. If you give the proof, I appreciate it. $$\sum_{k=1}^\infty \frac{1}{k^2}\left( 1 + \frac12 + \dots + \frac1k \right)^...
0
votes
0answers
17 views

Producing every digit from a radix through a sequence

To get this out of the way - I am not a mathematician, but I have encountered an interesting problem that I think is very obvious, thus it has definitely been expressed before, but I do not know how ...
-1
votes
0answers
38 views

Mathematical Series [on hold]

I was just going through some series and came across one which I wasn’t able to solve. The first 8 numbers of the series are: $$ 1,3,7,19,53,153,449,1331,... $$ Can anyone tell me a mathematical ...
0
votes
2answers
15 views

Proof of product identities.

The following Lemma is from P.M. Cohn: Algebra Volume 1 (2nd ed.), 1982 Given $a, b_1, ..., b_r\in\Bbb Z$ if $a$ is coprime to each of $b_1,...,b_r$, then $a$ is coprime to $b_1...b_r$. Proof. ...
49
votes
7answers
8k views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^32^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
2
votes
5answers
67 views

Show that $ \lim_{n\to\infty}[\frac{1}{\sqrt n}+\frac{1}{\sqrt {n+1}}+\frac{1}{\sqrt {n+2}}…\frac{1}{\sqrt {2n}}] = \infty$

Show that $$ \lim_{n\to\infty}[\frac{1}{\sqrt n}+\frac{1}{\sqrt {n+1}}+\frac{1}{\sqrt {n+2}}.......\frac{1}{\sqrt {2n}}] = \infty$$ LHS : $ \lim_{n\to\infty}\frac{1}{n}[\frac{n}{\sqrt n}+\frac{n}{\...
9
votes
5answers
340 views

Prove $\lim\limits_{n\to \infty}\frac{1}{\sqrt n}\left|\sum\limits_{k=1}^n (-1)^k\sqrt k\right|= \frac{1}{2}$

I'm trying to show that $$\lim_{n\to \infty} x_n=\lim_{n\to \infty}\frac{1}{\sqrt n}\left|\sum_{k=1}^n (-1)^k\sqrt k\right|= \frac{1}{2}.$$ Assuming $\lim\limits_{n\to\infty} x_n=x$ exists, we have $$...
2
votes
1answer
92 views

Does some Lucas sequence contain infinitely many primes?

Does some nontrivial Lucas sequence contain infinitely many primes? The Mersenne numbers $M_n=2^n-1:n$ not necessarily prime are a Lucas sequence with recurrence relation $x_{n+1}=2x_n+1$. It's an ...
3
votes
3answers
63 views

Evaluate $\frac{9}{1!}+\frac{19}{2!}+\frac{35}{3!}+\frac{57}{4!}+\frac{85}{5!}+…$

Prove that $$\frac{9}{1!}+\frac{19}{2!}+\frac{35}{3!}+\frac{57}{4!}+\frac{85}{5!}+......=12e-5$$ $$ e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+...... $$ I have no clue of ...
10
votes
2answers
777 views

Applications of equidistribution

What are applications of equidistributed sequences ? I'm looking for examples of problems/questions in fields (not necessarily mathematical) where equidistribution is an ad-hoc tool towards a ...
0
votes
1answer
36 views

Perfect Squares in Fibonacci Numbers [duplicate]

Let us define Fibonacci Numbers as: $F_1=F_2=1,F_n=F_{n-1}+F_{n-2}$ for $n>2$. $F_1$,$F_2$, and $F_{12}$ are perfect squares. Find the least integer $n>12$, if any, such that $F_{n}$ is a ...
6
votes
3answers
92 views

proof that $\frac{a_{4n}-a_2}{a_{2n+1}}$ : integer

I would appreciate if somebody could help me with the following problem: Q: How to proof? If $\{a_n\}$ satisfy $a_{1}=a$, $a_2=b$, $a_{n+2}=a_{n+1}+a_{n}$($a,b$: positive integers) then proof ...
15
votes
2answers
371 views

A limit related to asymptotic growth of tetration

The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence $$ {^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
3
votes
1answer
49 views

Con. func. that for every sequence $(x_n) \in [-1,1]^{\mathbb Z}$ there is $t \in \mathbb R$ satisfying $f(t+n) = x_n$ for all $n$

Does there exist a continuous function $f : \mathbb R \rightarrow \mathbb R$ such that for every sequence $(x_n) \in [-1,1]^{\mathbb Z}$ there is $t \in \mathbb R$ satisfying $f(t+n) = x_n$ for all $n$...
3
votes
1answer
62 views

Please help me understand the following transition in the limit

I was trying to figure out how a limit was calculated and got stuck when trying to understand one of the proposed solutions: (note that this is just a small part of the solution, but the one that got ...
23
votes
9answers
384 views

List of integrals or series for Gieseking's constant $\rm{Cl}_2\big(\tfrac{\pi}3\big)$?

Catalan's constant $K$ can be defined as, $$K = \text{Cl}_2\big(\tfrac{\pi}2\big) = \Im\, \rm{Li}_2\big(e^{\pi i/2}\big)= \sum_{n=0}^\infty\left(\frac1{(4n+1)^2}-\frac1{(4n+3)^2}\right)=0.91596\dots$$ ...
1
vote
0answers
16 views

Theorem about monotone function's continuity

$$f: Df \rightarrow R, Df \subset R$$ f is monotone function and $Rf$ is interval. Then $f$ is continuous. Proof: Suppose that $$\exists \epsilon >0 : \forall \delta >0 \, \, \exists x\in (...
3
votes
3answers
75 views

Find $ \frac{1}{2^2 –1} + \frac{1}{4^2 –1} + \frac{1}{6^2 –1} + \ldots + \frac{1}{20^2 –1} $

Find the following sum $$ \frac{1}{2^2 –1} + \frac{1}{4^2 –1} + \frac{1}{6^2 –1} + \ldots + \frac{1}{20^2 –1} $$ I am not able to find any short trick for it. Is there any short trick or do we ...
1
vote
0answers
17 views

Generalization of Specialized Card Sort

Problem: Given $n$ unique cards in a series from 1 to $n$ inclusive, arrange the cards such that drawing the first card, then placing the next card at the back of the deck, then drawing the next card ...
3
votes
0answers
25 views

Series of product of Bessel functions

The Christoffel-Darboux formula applied to Bessel functions states that $$\sum\limits_{j=0}^{+\infty}J_{j+n}(t)J_{j+m}(t)=\frac{t}{2(m-n)}\left(J_{m-1}(t)J_n(t)-J_m(t)J_{n-1}(t) \right)$$ See for ...
0
votes
2answers
41 views

General formula for a SUM

I am currently working on a problem related to graph theory an I came across this sum. $ {\displaystyle a(n)=\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {n!}{2^{k}(n-2k)!k!}}.} $ Can someone tell me if ...
1
vote
1answer
29 views

$\sum_i \sum_j \sum_k (1 \le i \le j \le k \le n) ~~f_i f_j f_k $ in terms of $\sum f_i~, \sum f^2 _i ~\mbox{and}~ \sum f^3 _i $

We often encounter triple sums such as $$ S_1 = \sum_i \sum_j \sum_k (1 \le i \ne j \ne k \ne n) ~~f_i f_j f_k $$ $$ S_2 = \sum_i \sum_j \sum_k (1 \le i < j < k < n) ~~f_i f_j f_k $$ ...
6
votes
6answers
138 views

How to show that $\sum_{n=1}^{\infty}\frac{H_n}{n^2+n}=\frac{\pi^2}{6}$

Wolfram Alpha shows that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2+n}=\zeta(2)=\frac{\pi^2}{6}$$ I want to prove this. Attempt: I tried to treat this as a telescoping series: $$\begin{align} \sum_{n=1}^{\...
0
votes
1answer
24 views

Convergence of $\sum_{n=1}^\infty \sqrt{n + 1} \log(1/n + 1) \sin(1/n)$

What's the convergence of this series? Due to the fact that: $\sin(1/n) \sim (1/n)$ $\sqrt{n+1} \sim \sqrt{n}$ $\log(1/n + 1) \sim e^{1/n}$ I think my general term is equivalent to $e^{1/n}/n^{1/2}$...
2
votes
2answers
30 views

Sum of infinite series with variable range between -1 to 1

Let $S$ denote the infinite sum $2 + 5x + 9x^2 + 14x^3 + 20x^4 + ....$, where $| x | < 1$, then what is the value of $S$? I am not able to find the generalized form. What is the trick to solve ...
0
votes
1answer
74 views

Find $\sum_{k=1}^n\frac{\left(H_k^{(p)}\right)^2}{k^p}$

Find $$\sum_{k=1}^n\frac{\left(H_k^{(p)}\right)^2}{k^p}\,,$$ where $H_k^{(p)}=1+\frac1{2^p}+\cdots+\frac1{k^p}$ is the $k$th generalized harmonic number of order $p$. Cornel proved in his book, (...
1
vote
0answers
129 views
+50

Find the vertical asymptotes of the graph $(\!{\rm C}\!)$ of the function $y= \frac{1}{x\,\sin\frac{1}{x}}$ .

Problem. Find the vertical asymptotes of the graph $(\!{\rm C}\!)$ of the function $y= \dfrac{1}{x\,\sin\frac{1}{x}}$ . Solution. All the solutions of the equation $x\,\sin\dfrac{1}{x}= 0$ are $x= 0$ ...
1
vote
1answer
37 views

Evaluating a series of Gaussians and Sines

I have derived an equation that includes the following sum: $$ \sum_{n=1}^\infty n \exp\left(-an^2\right) \sin\left(\frac{n \pi x }{L}\right). $$ Is there a way to figure out what function $f(x)$ this ...
2
votes
1answer
64 views

The sum $\sum_{k=0}^{n} (-1)^k \frac{{n \choose k}}{{n+k \choose k}}=\frac{1}{2}$ [duplicate]

The sum $$\sum_{k=0}^{n} (-1)^k \frac{{n \choose k}}{{n+k \choose k}}=\frac{1}{2}?$$ can be checked by Mathematica. Here, the question is how to do it by hand?
2
votes
1answer
56 views

How to I find a formula for $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \cdots + \frac{1}{n(n+1)}$ using the given method?

The following is problem $6$ $(iii)$ from chapter $2$ of Spivak's Calculus: The formula for $1^{2} + \cdots + n^{2}$ may be derived as follows. We begin with the formula $$ (k+1)^{3} - k^{3} = 3k^{...
4
votes
3answers
85 views

injectivity and surjectivity of a recursive function

Let $f:\mathbb{N} \to \mathbb{Q}^+$ defined as follows : $$\begin{cases} f(0) = 0 \\ f(2n) = \dfrac{1}{f(n)+1} \\ f(2n+1) = f(n)+1 \end{cases}$$ It is asked to prove the injectivity then the ...
0
votes
2answers
57 views

Evaluating $\sum_{n\ge 0}\frac1{4^n(n+1)}\binom{2n}n$ [on hold]

We have $$\arcsin 1 =\sum_{n\ge 0}\frac1{4^n(2n+1)}\binom{2n}n$$ I want to evaluate a similar sum, namely $$\sum_{n\ge 0}\frac1{4^n(n+1)}\binom{2n}n$$ Is there an expression for this?
84
votes
30answers
175k views

Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. How? What's the proof? Or maybe it is self apparent just looking at the above? PS: This problem is known as "The sum of the first $n$ positive ...
0
votes
1answer
132 views

Find the term of the G.P.

Find the term of the G.P. $1, 1.2, 1.44, 1.728,... $ which is just greater than $100$. Please somebody explain me the question? All i know is $a=1$ and $r=1.2$ Help :( should it be $T_{n} > 100$?...
1
vote
1answer
48 views

Find the first $3$ terms of the two possible geometric progressions.

The fourth term of a G.P is $3$ and the sixth term is $147$. Find the first $3$ terms of the two possible geometric progressions. Can you help me find $a$ and $r$? It is too complicated. I took two ...
1
vote
2answers
6k views

Geometric progression of 1 and 1/3tan^2θ

The first two terms of a geometric progression are where 0<θ<π/2 (i) Find the set of values of θ for which the progression is convergent. [2] What does convergent mean and how to solve this ?
0
votes
3answers
28 views

Trouble with finding geometric progression pattern

I have this system: $$ b_2-b_1 = 18 $$ $$ b_4-b_3 = 162 $$ I have to find $b_1$ (the first element) and $q$ (common ratio). Any ideas how to solve it?
6
votes
2answers
144 views

The summation $\sum_{k=0}^{2n} (-1)^k \frac{{4n \choose 2k}}{{2n \choose k}}=\frac{1}{1-2n}$

This summation has been created by perseverance over a long period of time by using the results from the Table of Series and Integrals By I.S. Gradshteyn and I.M. Rhyzik [GR]. The idea was to make ...
13
votes
2answers
350 views

On the general form of the family $\sum_{n=1}^\infty \frac{n^{k}}{e^{2n\pi}-1} $

I. $k=4n+3.\;$ From this post, one knows that $$\sum_{n=1}^\infty \frac{n^{3}}{e^{2n\pi}-1} = \frac{\Gamma\big(\tfrac{1}{4}\big)^8}{2^{10}\cdot5\,\pi^6}-\frac{1}{240}$$ and a Mathematica session ...
1
vote
2answers
32 views

If $T:H \to H$ is compact and $\{h_n\}$ is bounded, is $\{T(h_n)\}$ a compact subset of $H$?

If $T:H \to H$ is linear and bounded and compact and $\{h_n\}$ is bounded, is $\{T(h_n)\}$ a compact subset of $H$? We have $H$ as a Hilbert space. I am getting a problem with definition of compact ...
4
votes
1answer
220 views

Role of binomial coefficents in nested summations in layman terms

I has this doubt from almost two years and not getting a simple solution in layman terms. In short, the doubt is the about relation between binomial coefficients and the nesting summation. Recently I ...
-1
votes
1answer
31 views

Divergent alternating series with subsequences approaching to $+\infty,-\infty$

I'd like to discuss the following problem, Given the following series $$\sum\limits_{n=1}^{\infty} (e^{n\cdot cos(\pi n)} - n)$$ discuss the convergence,divergence... The question is how to ...
0
votes
1answer
41 views

If a real-valued sequence of functions is uniformly bounded , then it is pointwise bounded.

Show that if $\{f_n\}$ is a uniformly bounded real-valued sequence, then it is pointwise bounded. It seems quite obvious that $\mid f_n (x)\mid \leq M, \forall n,x$ means that for every $x, \mid f_n(...
2
votes
1answer
38 views

Rigorous proof of Bolzano-Weirstrass theorem for sequences

Revisiting some elementary proofs I'm trying to write them as rigorously as posible so I've tried it for Bolzano-Weierstrass' theorem for sequences, and it would be really helpful if you could tell me ...
0
votes
1answer
114 views

Summation of Recurrence relation

We have a function defined as : $h(x+2)$= $ 2h(x+1)$ + $2-h(x)$ when x is even , $h(x+2)$=$ 3h(x)$ when x is odd. Given two numbers $a,b$ where $a\le b$ We need to find $$\sum_{n=a}^{b} h(n)$$...
0
votes
1answer
28 views

Studying the convergence of $f_n(z)=\frac{\cos(\sqrt{nz})}{\sqrt{n+2z}}$

Study the convergence of the following sequence $f_n(z)=\frac{\cos(\sqrt{nz})}{\sqrt{n+2z}}$ for $z\in\mathbb{C}$. $\lim_{n\to\infty}\frac{\cos(\sqrt{nz})}{\sqrt{n+2z}}=0$ Then: $|\frac{\cos(\sqrt{...
1
vote
1answer
44 views

Convergence in $l^{p}$

Prove that, given $q \in [1,\infty] $, then $ l^{p} \hookrightarrow l^{q} $ for all $p \in [1, q]$. Consider the sequence $ x^{(n)}=\bigl( x^{(n)}_{k} \bigr)_{k \in \mathbb{N}_{0} } \ $ defined by \...
-2
votes
6answers
173 views

Convergenge or divergence of $ \sum_{n=1}^\infty e^{-n^{2}}$

I can't find the convergence or divergence of the following series by using aspect ratio test or comparison test. The series is: $$ \sum_{n=1}^\infty e^{-n^{2}}$$ Thanks.
6
votes
3answers
10k views

Find $\lim_{n \to \infty} \sqrt[n]{n!}$.

I am playing around with the root/ratio test to practice with series. I just showed that $\sum \frac{1}{n!}$ converges by using the ratio test. I decided to see how things would go with the root test ...
0
votes
3answers
38 views

I can't figure if these 2 series are convergent

I have 2 series. The first one is: $$\sum_{n=1}^\infty\frac{ln(n)}{n (n+1)^\frac{1}{2}}$$ I've tried to compare it with some kind of harmonic series but I can't find the ideal one. My problem is ...