Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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Different sequance problem

Let $x \in \Bbb{R}^+$, $a_1=x, a_2=x^{x}$ and $a_n=x^{a_{n-1}} \ \forall n\geq 3 $ $$S=\{x \in \Bbb{R}^+: a_n \rightarrow a \in \Bbb{R} \}$$ find the $sup(S)$ I can show $ 0<x<1 \vee 1<x<\...
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If Sigma(xn) and Sigma(yn) are convergent, show that Sigma(xn +yn) is convergent.

this is from exercise 3.7 introduction to real analysis bartle fourth edition number 4 i already tried to proof the Xn+Yn as a convergent but somehow it doesnt turn to proofed and still got some stuck
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1 vote
1 answer
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How to show that $\frac{\pi}{2} \le \sum_{n=0}^\infty \frac{1}{n^2+1} \le \frac{3\pi}{4}$

How to show that $\frac{\pi}{2} \le \sum_{n=0}^\infty \frac{1}{n^2+1} \le \frac{3\pi}{4}$ ? My Attempt : I was using Integral Test of a Series. I got $\int_0^\infty \frac{1}{1+x^2} \le \sum_{n=0}^\...
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Euler sum symmetry formula

The following identity $$\sum_{n=1}^{\infty} \frac{\mathcal{H}_n^{(p)}}{n^q} + \sum_{n=1}^{\infty} \frac{\mathcal{H}_n^{(q)}}{n^p} = \frac{\zeta(p+q)+ \zeta(q) \zeta(p)}{2}$$ is considered obvious. ...
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Rewriting the sum of a geometric series as a function

I have a question about geometric series: The task wants you to find a function $S(x)$ for the sum of a geometric sequence: $$2\sum_{n=1}^\infty (x-1)^{2n-1} $$ My first thought is to use: $$\sum_{n=...
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2 votes
0 answers
29 views

Help with finding analytical formula of a recurrence relation using iteration.

$$\displaystyle{\displaylines{g(0)=1}}$$ $$\displaystyle{\displaylines{g(n)=3^n-g(n-1)+1}}$$ Find the analytical definition of the recurrence relation using iteration $$\displaystyle{\displaylines{g_{...
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6 votes
2 answers
694 views

Longest geometric progression of primes

There are arbitrarily long arithmetic progressions of primes e.g. $5, 11, 17, 23, 29$ for a $5$-length progression, but no (infinite) arithmetic sequence of primes with common difference $d\neq 0$, as ...
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1 vote
1 answer
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How many binary arrays that have no $3$ consecutive $1$'s are there? [duplicate]

Let $N$ be a positive integer. The function $f(N)$ indicates that how many of the binary arrays of length $N$ don't consist of $3$ consecutive $1$'s. For example, if we'd have a look at $f(3)$: There ...
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Prove that $(k+2)^\alpha \sum\limits_{i=1}^{k}\frac{1}{(1+i)^\alpha} \ge (k+1)^\alpha \sum\limits_{i=1}^{k+1}\frac{1}{(1+i)^\alpha}$

In this problem, I'm trying to prove that for any $\alpha\in[1/2,1]$, there exists a constant $K(\alpha)$ such that for any $k\geqslant K(\alpha)$, the inequality $$(k+2)^\alpha\cdot\left(\frac{\sum_{...
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6 votes
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161 views

Let $2, 1+\frac{1}{2}, 3, 1+\frac{1}{3}, 4, 1+\frac {1}{4},\dots$ be a sequence. Does $a_n$ converge/diverge? Is there a $\sup$ or $\inf$?

Let $2,1+\frac{1}{2},3,1+\frac{1}{3},4,1+\frac {1}{4},...$ be a sequence then which of the statements is true? $a_n$ coverges to a finitie limit or diverges to infinity. $\limsup \limits_{n \to \...
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Closed form expression for discrete-time sum of a cosinusoid

I'm facing a problem in digital signal processing and am wondering if there is a closed form expression for the sum $$Y[n] = \sum_{k=0}^{n-1}\cos(\frac{2\pi k}{f}),$$ where $n$ < $f$. In case I ...
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Find a discontinuous linear transformation from the set of bounded real sequences to the reals

I am looking for a linear transformation T: B(N,R) -> R, with R the real numbers and B(N,R) the set of all bounded real sequences (with the sup norm), that is discontinuous. I first thought that ...
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1 vote
1 answer
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Prove that $\sum_{r=1}^{3k} r \tan(60r)^{\text{o}}=-k\sqrt{3}$

Prove that $$\sum_{r=1}^{3k} r \tan(60r)^{\text{o}}=-k\sqrt{3}$$ I did this by re-writing the sum as $$\sum_{r=1}^{3k} r \tan(60r)^{\text{o}}=-k\sqrt{3} = \sum_{r=1}^{k} \left((3r-2)\tan(60(3r-2))^{\...
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-1 votes
2 answers
48 views

Determine convergence of series with natural logarithm [closed]

I am trying to determine the convergence of the series below: $$\sum_{n=1}^{\infty}{\frac{(n+1)}{(n^2+2)\ln(n+3)}}$$ I've tried comparison test, Cauchy-condesation, but nothing seems to work.
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1 vote
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Mixing metric distance with Euclidean metric on $\mathbb{R}$

By far the most elegant proof of uniqueness of limits in a metric space I've seen (though I can't find the stack exchange link where it originated) is as follows. Let $X$ be a metric space, $(p_n)$ a ...
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-3 votes
1 answer
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Do Individual Sequences Converge if their Sum Converges? [closed]

Is the following statement true? If the sequence $\{a_n + b_n\}$ converges and $\{a_n\}$ also converges then so does $\{b_n\}$
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3 votes
3 answers
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Proof: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$?

Question: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$? Answer: Thank to @TonyK @Ryszard Szwarc. I think that i found an ever stronger demonstration that ...
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2 answers
78 views

How to show that the series $\sum_{n = 1}^{\infty} \frac{1}{n^n}$ converges

Show that $$ \sum_{n=1}^{\infty} \frac{1}{n^n}$$ I try to use the D'Alembert theorem bit I think to is not the good strategy, I think to the comparation test is the right way but, this is correct? $$\...
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3 answers
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How do we prove the limit of a sequence of real numbers is unique?

I wasn't sure if this proof was correct or not. Proposition. If $(a_n) \to a$ and $(a_n) \to b$, then $a = b$. Proof. Suppose $(a_n) \to a$ and $(a_n) \to b$. Then, $\lim \limits_{n \to \infty}(a_n)=a$...
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How do you come to the conclusion $0<1/(m+1)!+\dots+1/n!<1/(2^n)$ from $2^k<k!$ if $k\geq 4$?

I'm working out of the Bartle and Sherbert Introduction to Real Analysis book and I'm looking at the partial solution to proving the sequence $(1+1/2!+\dots+1/n!)$ is Cauchy (Exercise 3.5.2(b)). Their ...
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1 vote
0 answers
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Evaluate $\sum_{n=1}^\infty (\frac{1}{2}\sin x)^n $

I'm trying to answer this having been sent a photo of the questions from someone doing A level Edexcel Maths and I'm struggling. Any advice would be appreciated. Evaluate and justify the validity of ...
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3 votes
2 answers
114 views

Find $\sum_{k=1}^\infty\frac{1}{x_k^2-1}$ where $x_1=2$ and $x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}{2}$ for $n \ge 2$

Given $x_1=2$ and $x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}{2}, n\geq 2$ Prove that $y_n=\sum_{k=1}^{n}\frac{1}{x_k^2-1}, n\geq 1$ converges and find its limit. To prove a convergence we can just ...
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5 votes
3 answers
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Prove that $\sum_{n=0}^\infty \frac{1}{(2n+1)^2}=\frac{\pi^2}{8}$

I am asked to prove that $$\sum_{n=0}^\infty \frac{1}{(2n+1)^2}=\frac{\pi^2}{8}.$$ However, I am asked to prove it using the fact that $$\frac{\pi}{2}\tan\left(\frac{\pi}{2}z\right)=\sum_{m \text{ odd}...
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-4 votes
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Let $0 < a_1 < a_2$. Consider the sequence $a_n$ defined by $a_{n+1}$ = $(a_n + a_{n−1})/2$. Show that lim $a_n$ = $(a_1 + 2a_2)/3$ [duplicate]

Let $0 < a_1 < a_2$. Consider the sequence $a_n$ defined by $a_{n+1} = \frac{a_n + a_{n−1}}{2}$. Show that $\lim _{n \to \infty}a_n = \frac{a_1 + 2a_2}{3}$. I have tried substituting multiple ...
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Prove: in geometric sequence ($0\ <\ r\ <\ 1$) the ratio between a term and the sum of all following terms doesn't depend on the location of that term

Another question from my math finals, this time we were working with a geometric sequence. We were asked several questions about a specific sequence, but then the last question was this: prove that in ...
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-3 votes
0 answers
19 views

Prove a cauchy sequences [closed]

How to show that images problem is it true that the sequences convergen to x/y and that means the sequences is cauchy
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How can I show a simple inequality? [duplicate]

I was looking for some paper, and I had a question about simple inequality process. How can we show that following inequality? $\sum_{t=1}^{T}\frac{1}{t}\leq 1+\log{T}$ For those who want a detailed ...
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Is the following recursive sequence $a_n$ positive definite?

Is the following recursive sequence $a_n$ positive definite? $a_1=1\hspace{25pt}a_{n+1}= 2a_n +1$ I've been asked to prove also that's strictly increasing, both things by induction. Considering that ...
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0 votes
2 answers
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Transforming a double sum into a product of two sums.

Question is from Stein-Shakarchi Vol. I. I am asked to show that for two complex numbers $z_1$ and $z_2$, $e^{z_1}e^{z_2}=e^{z_1+z_2}$ using the definition of the complex exponential $e^z=\sum_{n=0}^{\...
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1 vote
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Is it true $\sum\limits_{i=1}^n \sum\limits_{j=n-i+1}^n i^{-a}j^{-a} \ge \int_1^n x^{-a}(\int_{n-x+1}^n y^{-a}\, \mathrm{d}y)\, \mathrm{d} x$?

Conjecture 1: For each $n\ge 1$ and $a \ge 2$, $$\sum_{i=1}^n \sum_{j=n-i+1}^n i^{-a}j^{-a} \ge \int_1^n x^{-a}\left(\int_{n-x+1}^n y^{-a}\, \mathrm{d}y\right) \mathrm{d} x.$$ This is related to this ...
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1 answer
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Is there a proof for the following Resummation formula?

While reading the paper (https://iopscience.iop.org/article/10.1088/1464-4266/5/3/363/pdf) I encountered the following resummation formula: $$ \sum_{n=0}^{\infty}\sum_{r=0}^{n}a_{n,r}=\sum_{n=0}^{\...
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1 vote
2 answers
63 views

Series of powers

Consider the sequences $a_k,b_k\in\mathbb R$ and let $N\in\mathbb N$. Do series of the form $$f(n)=\sum_{k=1}^N a_k\,{b_k}^n$$ have a name? They seem quite a natural object to study but I'm not sure ...
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1 vote
1 answer
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Does $\left\lbrace 1- \frac{1}{5-n^2} : n \in \Bbb N\right\rbrace$ admit a maximum and minimum?

Does the following set admit a maximum and minimum? $$A= \left\lbrace 1- \frac{1}{5-n^2} : n \in \Bbb N\right\rbrace$$ First, I've plot some of the set elements $$A = \left\lbrace \frac{3}{4},0, \...
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3 votes
2 answers
86 views

Proving that $a_i,b_i \to 0$ where $a_{i+1} := |b_i - a_i|$ and $b_{i+1} := |a_i - a_{i+1}|$?

Let $a_0 := 1$ and $b_0 := \sqrt{2}$, and define \begin{align*} a_{i+1} &:= |b_i - a_i| \\ b_{i+1} &:= |a_i - a_{i+1}| \end{align*} Prove that $\lim_{i \to \infty}{a_i} = 0$ and $\lim_{i \to \...
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0 answers
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Coinciding Limsup and limit

Let $(x_n)_{n \geq 1}$ be a sequence of real numbers. Suppose that we are able to show that for a fixed number $m$, $(y_n)_{n \geq 1}:= (x_{n+m})$ and we know that $\lim_{n\to\infty}(y_n)=x$ for some ...
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1 vote
1 answer
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Proof verification that $\lim_{n \to \infty}\frac{n!}{n^n}=0$ (sequence)

Using an epsilon-N approach (since this is supposed to be a sequence), we require $$\forall \varepsilon>0, \exists N \hspace{1mm}\text{s.t} \hspace{2mm}n>N \implies |a_n-L|<\varepsilon$$ Now, ...
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Harmonic Distribution of prime numbers [closed]

I developed a sieve that depicts the distribution of prime numbers as contained in harmonic (repetitive) patterns. Published it here What would be the process to know if I’m rightfully thinking this ...
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0 votes
1 answer
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How to bound $\sum e^{-a n^p}$

Let $0<p<1$ and $a>0$. Then it would seem that $$ \sum_1^\infty e^{-an^p}\le Ce^{-a} $$ For some constant $C(p)$ since the terms in the summation decay exponentially. However, I can't quite ...
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4 votes
1 answer
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A combined arithmetic and geometric sequence question

Here is a question I am currently struggling with - The first, the tenth and the twentieth terms of an increasing arithmetic sequence are also consecutive terms in an increasing geometric sequence. ...
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Decomposition of a summation to a product into smaller summations

I came across a proof that used the fact that $$\sum_{p=1}^{mn}p = \sum _{i=1}^{m}\sum _{j=1}^{n}(n(i-1)+j).$$ In general, I found that $$\sum_{p=1}^{\Pi_{k=1}^{n}{m_k}}p = \sum _{i_1=1}^{m_1}\sum _{...
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0 votes
3 answers
63 views

What is sum of $\sum^{\infty}_{k=3} \frac{q^k}{k}$? [closed]

what is the sum of $\sum^{\infty}_{k=3} \frac{q^k}{k}$, where $q \in (0,1)$?
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3 votes
0 answers
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Is Wolfram Alpha giving me a wrong answer?

I have asked for the convergence of the series $\sum (3^n/\sqrt{n})x^{2n+1}$, which has the radius of convergence of $1/\sqrt{3}$ and diverges at $|x|=1/\sqrt{3}$. However, the Wolfram Alpha is ...
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1 vote
2 answers
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Convergence of $ \sum_{n=1}^{\infty} \frac{(1+nx)^n}{n!} , x>0$

I have been trying this question $\frac{(1+nx)^n}{n!} > \frac{(nx)^n}{n!}$ Since $\sum_{n=1}^{\infty}\frac{(nx)^n}{n!}$ is divergent when $x \geq \frac{1}{e} \implies \sum_{n=1}^{\infty} \frac{(...
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0 votes
1 answer
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(Hint request) $n^{1/n^{1/3}}$ converges.

I can show that $n^{1/n}$ converges to $1$ by using binomial theorem to deduce that $n^{1/n}-1$ converges to $0$. However, similar method does not, at least directly, apply to show that $n^{1/{n^{1/3}}...
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0 votes
0 answers
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Show that an analytic function is always negative

I want to show the following function is negative for $z\in [0,1)$: $$f(z) = -1 + z^2(z-1) + 2\sum_{k=0}^\infty (-1)^k z^{(2k+1)^2+1}. $$ By Tauberian theorem, I know that $\lim_{z\to 1^-}f(z)=0$. We ...
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1 vote
0 answers
24 views

Calculate the expression from an infinitely differentiable function

Calculate the expression from an infinitely differentiable function: enter image description here
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7 votes
3 answers
184 views

How to evaluate the sum of $\sum_{n=0}^{\infty}\frac{1}{3n^{2}+4n+1}$

I hava an infinite sum $$\sum_{n=0}^{\infty}\frac{1}{3n^{2}+4n+1}$$ I factored the denominator $$\sum_{n=0}^{\infty}\frac{1}{\left(3n+1\right)\left(n+1\right)}$$ Then I separated the fraction $$\frac{...
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4 votes
3 answers
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Prove $\sum \frac{n^{p+1}}{a_1+2^pa_2+\cdots+n^pa_n}$ is convergent.

Assume $\sum\limits_{n=1}^{\infty} \dfrac{1}{a_n}$ is a convergent positive term series and $p>0$. Prove $$ \sum_{n=1}^{\infty} \frac{n^{p+1}}{a_1+2^pa_2+\cdots+n^pa_n}$$ is convergent. Since $$...
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0 votes
1 answer
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Is there an alternating series in which the sum is also one of the terms? [closed]

The geometric series 1/2 - 1/4 + 1/8 - 1/16 + ... sums to 1/3. But 1/3 is not one of the terms that appears in the series. Indeed, in general one would not expect the sum to appear as one of the terms....
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0 votes
2 answers
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Proof that $\sum_{n=1}^\infty\frac{F_n}{3^n n} = \frac{\ln(\phi+1)}{\sqrt{5}}$

I conjectured by computation the following, but I’m not sure where to start to prove it. $$\sum_{n=1}^\infty\frac{F_n}{3^n n} = \frac{\ln(\phi+1)}{\sqrt{5}}$$ where $F_n$ are the Fibonacci numbers. I’...
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