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

We often encounter triple sums such as $$S_1 = \sum_i \sum_j \sum_k (1 \le i \ne j \ne k \ne n) ~~f_i f_j f_k$$ $$S_2 = \sum_i \sum_j \sum_k (1 \le i < j < k < n) ~~f_i f_j f_k$$ ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 $T_{n} > 100$?...
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### 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 ...
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### 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 ?
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 \...
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### 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 ...