# Tagged Questions

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

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### Are all sets of $n$, s.t. $R(m)^n=I$, where $R(m)$ is any sequence of $m$ moves on a Rubik's cube and $I$ is the identity operator, known?

I've written a program that finds the number of times, $n$, one must apply any operation $R_i(m)$, which consists of $m$ single moves/turns/elementary operations on a Rubik's cube, s.t. $R_i(m)^n=I$, ...
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### Convergence of a Harmonic Continued Fraction

Does this continued fraction converge? $$\large\frac { 1 }{ 1+\frac { 1 }{ 2+\frac { 1 }{ 3+\frac { 1 }{ 4+\dots } } } }$$ ($[0;1,2,3,4, \dots]$) I tried approximating a few values but I ...
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### Show $\sum_{k=1}^{n}\frac{1}{k}\sim \ln(n)$

there is an example of how we apply Integral test for convergence Theorem: Consider an $n_{0}$ and a non-negative, continuous function $f$ defined on the unbounded $[n_0,+\infty[$, on which it ...
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### Limits of $a_n, b_n, c_n$

Three given positive $a_1, b_1, c_1$, such that $a_1+b_1+c_1=1, \forall\ n,$ $$a_{n+1}=a_n^2+2b_nc_n, b_{n+1}=b_n^2+2a_nc_n, c_{n+1}=c_n^2+2a_nb_n$$ Prove $\{a_n\},\{b_n\}$ and $\{c_n\}$ are ...
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### Having no derangements — any advantage?

Is there any problem-solving advantage when a sequence has no derangements? Edit Perhaps the non-derangements are just a by-product of the argument that we find primes in $(m-k,k] \iff k|m$? In an ...
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### Uniformly Cauchy sequence of functions

I am trying to show the following: For each $n \in \mathbb N$, let $f_n:X \to Y$, where $(Y,d)$ is a complete metric. Suppose that for every $\epsilon>0$, there exists $n_0 \in \mathbb N$ such ...
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### Prove that $\lim a_{n}=e$?

Given that $a_{1}=0$, $a_{2}=1$ and $$a_{n+2}=\frac{(n+2)a_{n+1}-a_{n}}{n+1}$$ prove that $\lim\limits_{n\to\infty} a_n=e$ What I did: It was hinted to prove that $a_{n+1}-a_{n}=\frac{1}{n!}$ ...
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### Coefficients in series expansion of $\left(\frac{x}{1-x}\right)^3$

How can I find the coefficient of $x^m$ in $$\left(\sum_{n=1}^{\infty}x^n\right)^3$$ for some $m\in{\Bbb N}$? I attempted it using the Cauchy products, but things got messy. I think one might want ...
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### How to determine when a rounded sequence “converges” and what the convergence value is

I have a process on a server that iteratively computes a value over time. The value follows a fairly simple formula, which would generate a convergent sequence; however, the value of the formula is ...
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### Why is this series of square root of twos equal $\pi$?

Wikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+\cdots+ \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, ...
### nature of series $\sum_{n\geq 0}(-1)^{n}u_n$
Let $(u_n)_{n\in\mathbb{N}}$ be sequence defined as follows: $$\left\{ \begin{cases} u_0\in\mathbb{R}^{+}\\\forall n\in\mathbb{N},\quad u_{n+1}=\dfrac{e^{-u_n}}{n+1}\\ \end{cases} \right\}$$ ...