# 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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### Evaluate $\lim_{n\to\infty}\frac{3}{n}\sum_{i=1}^n\left(\left(\frac{3i}{n}\right)^2-\left(\frac{3i}{n}\right)\right)$

How to evaluate the following? $$\lim_{n\to\infty}\frac{3}{n}\sum_{i=1}^n\left(\left(\frac{3i}{n}\right)^2-\left(\frac{3i}{n}\right)\right)$$ I simply expanded but I did not find the answer. I think ...
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### Problem concerning a question of sequence.

The question is : Let $S$ be an infinite subset of $\mathbb {R}$ that is bounded above and let $\sup S \notin S$. Show that there exists a monotone increasing sequence $\{u_n\} \subset S$ , such ...
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### Number of ways to connect vertices of n squares with line segments

What is the number of ways to connect the vertices of n squares with non-intersecting line segments ? These line segments should not cross the edges of the given squares as well. Obviously $N(3)$ is ...
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### $(a_n) \to L \Rightarrow (a_n + \alpha) \to L + \alpha$ [closed]

How can I proof: $(a_n)$ sequence, $\alpha, L \in \Bbb R$: $(a_n) \to L \Rightarrow (a_n + \alpha) \to L + \alpha$
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### Can anyone help me to endorse in ArXiv (math.GM) for my article (http://vixra.org/abs/1606.0034) in ViXra? [closed]

Recently, my article is in Vixra.org http://vixra.org/abs/1606.0034 but i want to publish this article in ArXiv(math.GM) to get more useful comments. I have found that it needs endorsement so i am ...
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### set-theoretic definition of sum of sequences, difference of sequences, product of sequences,…

I have $f,g \in \Bbb R^\Bbb N$, I define: $(f+g):=\{(y,z)| z= f(y)+g(y) \}$ $(-f):=\{(y,z)| z=-f(y) \}$ $(f-g):=f+(-g)$ $(f\cdot g):=\{(y,z)| z= f(y)\cdot g(y) \}$ Is it correct? I have problem ...
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### Is the partial sum of cosine bounded?

Is it true that $\sum_{k=1}^n \cos k$ is a bounded sequence? If so how to prove? I want to prove the series of $\cos n/(\sqrt n)$ is convergent by abel test but I dont know if the partial sums of ...
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### How to determine sum of an alternating power series and to prove that sum is positive

I am working on a problem involving an alternating power series as follows: $$\sum_{i=0}^{a-2} (-1)^{a+b-i-2}(a+b-i-1)x^{a+b-i-2}$$ $a$ and $b$ is constant with $0<x<1$ I would like to ...
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### Fractal dimension of the function $f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}$

Consider the function $$f(x)=\sum_{n=1}^{\infty}\frac{\mathrm{sign}\left(\sin(nx)\right)}{n}\, .$$ This is a bizarre and fascinating function. A few properties of this function that SEEM to be true: ...
Okay so I have a measurable function $f$ and a set $E_{n,i}$ $$E_{n,i}=\left\{ x:\frac{i-1}{2^{n}} \leq f(x)<\frac{i}{2^{n}}\right\}$$ where $i=1,...,n2^{n}$ and $n=1,2,...$ Then I have another ...