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.

44,500 questions
29 views

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 ...
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 ...
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 ...
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 ...
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 ...
371 views

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 ...
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 ...
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 ...
41 views

General formula for a SUM

I am currently working on a problem related to graph theory an I came across this sum. $a(n)=\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {n!}{2^{k}(n-2k)!k!}}.}$ Can someone tell me if ...
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 $$S_1 = \sum_i \sum_j \sum_k (1 \le i \ne j \ne k \ne n) ~~f_i f_j f_k$$ $$S_2 = \sum_i \sum_j \sum_k (1 \le i < j < k < n) ~~f_i f_j f_k$$ ...
138 views

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 ...
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?
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 ...
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 $T_{n} > 100$?...
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 ...
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.  What does convergent mean and how to solve this ?
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?
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 ...
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 ...
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 ...
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 ...
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 ...
41 views

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 \...
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.
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 ...
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 ...